This is the list of all generating functions, and the first , 41, terms in the enumerating series for P-partitions of staircase shapes (i.e. shifted-tableaux with the same number of cells in each row and the rows are jutting-to the left or right ) of up to, 6, levels with the number of cells in each row <=, 5 The Two-rowed, atomic object with, 2, cells in each row and the second row is jutting by, -1, cells compared to the one above, which in our notation is the poset [{}, {1}, {}, {1, 3}] The generating functions for up to, 6, rows i.e. up to the, 5, -th power of it are as follows This is the story of the P-partitions for the n-th power of the poset [{}, {1}, {}, {1, 3}] glued by the first and last, 2, vetrices for n from 1 to, 5 as well as the first , 41, terms of its counting series For the, 1, power which is the poset [{}, {1}, {}, {1, 3}] the generating function for its P-partitions is (q^4+q^3+q+q^2+1)/(q-1)/(q^2-1)/(q^3-1)/(q^4-1) The first, 41, coefficients are [1, 2, 4, 7, 12, 17, 25, 34, 46, 59, 76, 94, 117, 141, 170, 201, 238, 276, 321, 368, 422, 478, 542, 608, 683, 760, 846, 935, 1034, 1135, 1247, 1362, 1488, 1617, 1758, 1902, 2059, 2219, 2392, 2569, 2760] For the, 2, power which is the poset [{}, {1}, {}, {1, 3}, {}, {3, 5}] the generating function for its P-partitions is (q^12+2*q^11+3*q^10+5*q^9+7*q^8+8*q^7+9*q^6+8*q^5+7*q^4+5*q^3+3*q^2+2*q+1)/(q-1 )/(q^2-1)/(q^3-1)/(q^6-1)/(q^5-1)/(q^4-1) The first, 41, coefficients are [1, 3, 7, 15, 29, 51, 86, 136, 208, 307, 441, 617, 847, 1138, 1506, 1962, 2524, 3205, 4029, 5010, 6177, 7549, 9157, 11024, 13189, 15675, 18528, 21777, 25471, 29644, 34354, 39636, 45556, 52156, 59506, 67654, 76681, 86635, 97607, 109654, 122871] For the, 3, power which is the poset [{}, {1}, {}, {1, 3}, {}, {3, 5}, {}, {5, 7}] the generating function for its P-partitions is (1+3*q+86*q^8+67*q^7+q^24+103*q^15+6*q^22+21*q^20+33*q^19+128*q^13+67*q^17+86*q ^16+118*q^10+49*q^6+128*q^11+6*q^2+12*q^3+21*q^4+103*q^9+33*q^5+131*q^12+49*q^ 18+3*q^23+12*q^21+118*q^14)/(q^5-1)/(q^6-1)/(q^7-1)/(q^8-1)/(q-1)^2/(q^3-1)/(q^ 4-1)/(q+1) The first, 41, coefficients are [1, 4, 11, 27, 59, 118, 222, 395, 672, 1099, 1740, 2676, 4013, 5884, 8456, 11934, 16572, 22671, 30600, 40791, 53761, 70110, 90546, 115881, 147062, 185165, 231432, 287263, 354260, 434213, 529156, 641350, 773341, 927944, 1108313, 1317914, 1560608, 1840620, 2162631, 2531743, 2953583] For the, 4, power which is the poset [{}, {1}, {}, {1, 3}, {}, {3, 5}, {}, {5, 7}, {}, {7, 9}] the generating function for its P-partitions is (1+4*q+10*q^38+246*q^33+781*q^30+47*q^36+87*q^35+23*q^37+379*q^8+246*q^7+2778*q ^24+2439*q^15+3297*q^22+q^40+151*q^34+3489*q^20+3440*q^19+1713*q^13+3070*q^17+ 2778*q^16+781*q^10+151*q^6+4*q^39+1052*q^11+10*q^2+23*q^3+47*q^4+556*q^9+87*q^5 +1366*q^12+3297*q^18+3070*q^23+3440*q^21+1713*q^27+556*q^31+379*q^32+2439*q^25+ 1052*q^29+2076*q^14+2076*q^26+1366*q^28)/(q^10-1)/(q^8-1)/(q^9-1)/(q^6-1)/(q^7-\ 1)/(q-1)/(q^2-1)/(q^3-1)/(q^5-1)/(q^4-1) The first, 41, coefficients are [1, 5, 16, 44, 107, 237, 490, 956, 1776, 3165, 5441, 9060, 14670, 23167, 35770, 54119, 80382, 117390, 168807, 239312, 334831, 462801, 632479, 855290, 1145253, 1519432, 1998482, 2607256, 3375492, 4338576, 5538438, 7024494, 8854756, 11097029, 13830262, 17146008, 21150106, 25964433, 31728928, 38603739, 46771625] For the, 5, power which is the poset [{}, {1}, {}, {1, 3}, {}, {3, 5}, {}, {5, 7}, {}, {7, 9}, {}, {9, 11}] the generating function for its P-partitions is (1+5*q+670*q^53+82781*q^38+133311*q^33+90*q^56+5*q^59+6919*q^48+13812*q^46+ 24747*q^44+4678*q^49+49727*q^41+143927*q^30+105696*q^36+116226*q^35+94403*q^37+ 1162*q^8+670*q^7+105696*q^24+18724*q^15+82781*q^22+60117*q^40+125565*q^34+9918* q^47+18724*q^45+3057*q^50+60117*q^20+49727*q^19+31936*q^43+1162*q^52+9918*q^13+ 31936*q^17+24747*q^16+3057*q^10+366*q^6+71233*q^39+188*q^55+15*q^58+1924*q^51+ 366*q^54+q^60+4678*q^11+15*q^2+39*q^3+90*q^4+1924*q^9+188*q^5+6919*q^12+40286*q ^18+94403*q^23+71233*q^21+39*q^57+133311*q^27+142709*q^31+139114*q^32+40286*q^ 42+116226*q^25+142709*q^29+13812*q^14+125565*q^26+139114*q^28)/(q^9-1)/(q^10-1) /(q^11-1)/(q^12-1)/(q^7-1)/(q^8-1)/(q-1)^3/(q^5-1)/(q^6-1)/(q^3-1)/(q^3+q^2+q+1 )/(q+1) The first, 41, coefficients are [1, 6, 22, 67, 179, 433, 972, 2052, 4113, 7888, 14558, 25973, 44966, 75777, 124631, 200508, 316150, 489375, 744762, 1115794, 1647560, 2400130, 3452737, 4908921, 6902822, 9606801, 13240656, 18082663, 24482753, 32878160, 43811911, 57954560, 76129666, 99343490, 128819514, 166038400, 212784099, 271196839, 343833910, 433739072, 544521673] The whole thing took, 3.820, seconds of CPU time The Two-rowed, atomic object with, 2, cells in each row and the second row is jutting by, 0, cells compared to the one above, which in our notation is the poset [{}, {1}, {1}, {1, 2, 3}] The generating functions for up to, 6, rows i.e. up to the, 5, -th power of it are as follows This is the story of the P-partitions for the n-th power of the poset [{}, {1}, {1}, {1, 2, 3}] glued by the first and last, 2, vetrices for n from 1 to, 5 as well as the first , 41, terms of its counting series For the, 1, power which is the poset [{}, {1}, {1}, {1, 2, 3}] the generating function for its P-partitions is 1/(q^2-1)^2/(q^3-1)/(q-1) The first, 41, coefficients are [1, 1, 3, 4, 7, 9, 14, 17, 24, 29, 38, 45, 57, 66, 81, 93, 111, 126, 148, 166, 192, 214, 244, 270, 305, 335, 375, 410, 455, 495, 546, 591, 648, 699, 762, 819, 889, 952, 1029, 1099, 1183] For the, 2, power which is the poset [{}, {1}, {1}, {1, 2, 3}, {1, 3}, {1, 2, 3, 4, 5}] the generating function for its P-partitions is 1/(q^4-1)/(q^4+q^3-q-1)/(q-1)^2/(q^3-1)/(q^2-1) The first, 41, coefficients are [1, 1, 3, 5, 9, 13, 22, 30, 45, 61, 85, 111, 150, 190, 247, 309, 390, 478, 593, 715, 870, 1038, 1243, 1465, 1735, 2023, 2368, 2740, 3175, 3643, 4189, 4771, 5443, 6163, 6982, 7858, 8852, 9908, 11098, 12366, 13780] For the, 3, power which is the poset [{}, {1}, {1}, {1, 2, 3}, {1, 3}, {1, 2, 3, 4, 5}, {1, 3, 5}, {1, 2, 3, 4, 5, 6, 7}] the generating function for its P-partitions is 1/(q^5-1)/(q^4+q^3-q-1)/(q^4-1)^2/(q-1)^2/(q^3-1)/(q^2-1) The first, 41, coefficients are [1, 1, 3, 5, 10, 15, 26, 38, 60, 85, 125, 172, 243, 325, 442, 580, 767, 986, 1275, 1612, 2045, 2548, 3179, 3910, 4812, 5849, 7109, 8554, 10285, 12259, 14599, 17255, 20372, 23895, 27991, 32603, 37925, 43890, 50725, 58361, 67053 ] For the, 4, power which is the poset [{}, {1}, {1}, {1, 2, 3}, {1, 3}, {1, 2, 3, 4, 5}, {1, 3, 5}, {1, 2, 3, 4, 5, 6, 7}, {1, 3, 5, 7}, {1, 2, 3, 4, 5, 6, 7, 8, 9}] the generating function for its P-partitions is 1/(q^3-1)^3/(q^4-1)/(q^6+q^5-q-1)/(q-1)/(q^4-q^3+q-1)/(q^5-1)/(q^2-1)^2/(q^2+1) The first, 41, coefficients are [1, 1, 3, 5, 10, 16, 28, 42, 68, 100, 151, 215, 312, 432, 605, 821, 1117, 1485, 1977, 2581, 3371, 4335, 5566, 7060, 8938, 11196, 13994, 17338, 21426, 26280, 32152, 39074, 47369, 57093, 68637, 82097, 97955, 116339, 137849, 162665, 191507] For the, 5, power which is the poset [{}, {1}, {1}, {1, 2, 3}, {1, 3}, {1, 2, 3, 4, 5}, {1, 3, 5}, {1, 2, 3, 4, 5, 6, 7}, {1, 3, 5, 7}, {1, 2, 3, 4, 5, 6, 7, 8, 9}, {1, 3, 5, 7, 9}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}] the generating function for its P-partitions is 1/(q^3-1)^3/(q^5-1)/(q^6+q^5-q-1)/(q^8+q^6-q^2-1)/(q-1)/(q^4-1)/(q^7-1)/(q^4-q^ 3+q-1)/(q^2-1)^2 The first, 41, coefficients are [1, 1, 3, 5, 10, 16, 29, 44, 72, 108, 166, 241, 357, 504, 720, 998, 1386, 1882, 2559, 3413, 4551, 5981, 7842, 10162, 13138, 16811, 21454, 27150, 34251, 42898, 53570, 66464, 82221, 101146, 124057, 151404, 184261, 223235, 269723, 324578, 389560] The whole thing took, 1.452, seconds of CPU time The Two-rowed, atomic object with, 2, cells in each row and the second row is jutting by, 1, cells compared to the one above, which in our notation is the poset [{}, {1}, {1, 2}, {1, 2, 3}] The generating functions for up to, 6, rows i.e. up to the, 5, -th power of it are as follows This is the story of the P-partitions for the n-th power of the poset [{}, {1}, {1, 2}, {1, 2, 3}] glued by the first and last, 2, vetrices for n from 1 to, 5 as well as the first , 41, terms of its counting series For the, 1, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}] the generating function for its P-partitions is 1/(q-1)/(q^2-1)/(q^3-1)/(q^4-1) The first, 41, coefficients are [1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 27, 34, 39, 47, 54, 64, 72, 84, 94, 108, 120, 136, 150, 169, 185, 206, 225, 249, 270, 297, 321, 351, 378, 411, 441, 478, 511, 551, 588, 632] For the, 2, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}] the generating function for its P-partitions is 1/(q-1)/(q^2-1)/(q^3-1)/(q^4-1)/(q^5-1)/(q^6-1) The first, 41, coefficients are [1, 1, 2, 3, 5, 7, 11, 14, 20, 26, 35, 44, 58, 71, 90, 110, 136, 163, 199, 235, 282, 331, 391, 454, 532, 612, 709, 811, 931, 1057, 1206, 1360, 1540, 1729, 1945, 2172, 2432, 2702, 3009, 3331, 3692] For the, 3, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}] the generating function for its P-partitions is 1/(q-1)/(q^2-1)/(q^3-1)/(q^4-1)/(q^5-1)/(q^6-1)/(q^7-1)/(q^8-1) The first, 41, coefficients are [1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 40, 52, 70, 89, 116, 146, 186, 230, 288, 352, 434, 525, 638, 764, 919, 1090, 1297, 1527, 1801, 2104, 2462, 2857, 3319, 3828, 4417, 5066, 5812, 6630, 7564, 8588, 9749] For the, 4, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5, 6, 7, 8}, {1, 2, 3, 4, 5, 6, 7, 8, 9}] the generating function for its P-partitions is 1/(q-1)/(q^2-1)/(q^3-1)/(q^4-1)/(q^5-1)/(q^6-1)/(q^7-1)/(q^8-1)/(q^9-1)/(q^10-1 ) The first, 41, coefficients are [1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 55, 75, 97, 128, 164, 212, 267, 340, 423, 530, 653, 807, 984, 1204, 1455, 1761, 2112, 2534, 3015, 3590, 4242, 5013, 5888, 6912, 8070, 9418, 10936, 12690, 14663, 16928] For the, 5, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5, 6, 7, 8}, {1, 2, 3, 4, 5, 6, 7, 8, 9}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}] the generating function for its P-partitions is 1/(q-1)/(q^2-1)/(q^3-1)/(q^4-1)/(q^5-1)/(q^6-1)/(q^7-1)/(q^8-1)/(q^9-1)/(q^10-1 )/(q^11-1)/(q^12-1) The first, 41, coefficients are [1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 100, 133, 172, 224, 285, 366, 460, 582, 725, 905, 1116, 1380, 1686, 2063, 2503, 3036, 3655, 4401, 5262, 6290, 7476, 8877, 10489, 12384, 14552, 17084, 19978, 23334] The whole thing took, 0.884, seconds of CPU time The Two-rowed, atomic object with, 3, cells in each row and the second row is jutting by, -2, cells compared to the one above, which in our notation is the poset [{}, {1}, {1, 2}, {}, {4}, {1, 4, 5}] The generating functions for up to, 6, rows i.e. up to the, 5, -th power of it are as follows This is the story of the P-partitions for the n-th power of the poset [{}, {1}, {1, 2}, {}, {4}, {1, 4, 5}] glued by the first and last, 3, vetrices for n from 1 to, 5 as well as the first , 41, terms of its counting series For the, 1, power which is the poset [{}, {1}, {1, 2}, {}, {4}, {1, 4, 5}] the generating function for its P-partitions is (3*q^6+3*q^5+3*q^4+2*q^7+q^9+q^8+1+2*q^2+2*q^3+q)/(q-1)/(q^2-1)/(q^3-1)/(q^4-1) /(q^5-1)/(q^6-1) The first, 41, coefficients are [1, 2, 5, 9, 17, 28, 46, 69, 103, 147, 207, 282, 381, 501, 654, 838, 1065, 1334, 1660, 2040, 2493, 3017, 3631, 4334, 5150, 6075, 7137, 8334, 9694, 11217, 12936, 14847, 16989, 19360, 21999, 24906, 28126, 31655, 35544, 39792, 44450 ] For the, 2, power which is the poset [{}, {1}, {1, 2}, {}, {4}, {1, 4, 5}, {}, {7}, {4, 7, 8}] the generating function for its P-partitions is -(1+2*q+64*q^8+48*q^7+10*q^24+123*q^15+27*q^22+54*q^20+70*q^19+126*q^13+102*q^ 17+115*q^16+96*q^10+35*q^6+109*q^11+5*q^2+8*q^3+15*q^4+80*q^9+23*q^5+121*q^12+ 87*q^18+17*q^23+39*q^21+q^27+5*q^25+128*q^14+2*q^26)/(q-1)/(q^2-1)/(q^3-1)/(q^4 -1)/(q^5-1)/(q^9-1)/(q^6-1)/(q^7-1)/(q^8-1) The first, 41, coefficients are [1, 3, 9, 20, 44, 86, 162, 286, 489, 802, 1280, 1982, 3004, 4448, 6470, 9239, 12998, 18013, 24649, 33307, 44521, 58878, 77135, 100127, 128907, 164633, 208738, 262798, 328728, 408633, 505035, 620696, 758891, 923198, 1117809, 1347299, 1616979, 1932626, 2300894, 2729008, 3225221] For the, 3, power which is the poset [{}, {1}, {1, 2}, {}, {4}, {1, 4, 5}, {}, {7}, {4, 7, 8}, {}, {10}, {7, 10, 11} ] the generating function for its P-partitions is (1+3*q+3*q^53+5902*q^38+12875*q^33+166*q^48+457*q^46+1050*q^44+92*q^49+2822*q^ 41+16222*q^30+8586*q^36+10028*q^35+7198*q^37+384*q^8+236*q^7+15738*q^24+4191*q^ 15+13471*q^22+3700*q^40+11475*q^34+283*q^47+707*q^45+47*q^50+10692*q^20+9247*q^ 19+1507*q^43+9*q^52+2452*q^13+6505*q^17+5284*q^16+892*q^10+140*q^6+4731*q^39+22 *q^51+q^54+1287*q^11+9*q^2+19*q^3+41*q^4+595*q^9+77*q^5+1805*q^12+7840*q^18+ 14694*q^23+12115*q^21+17329*q^27+15302*q^31+14169*q^32+2092*q^42+16547*q^25+ 16885*q^29+3249*q^14+17088*q^26+17261*q^28)/(q^12-1)/(q^10-1)/(q^9-1)/(q^11-1)/ (q-1)/(q^2-1)/(q^3-1)/(q^4-1)/(q^5-1)/(q^8-1)/(q^6-1)/(q^7-1) The first, 41, coefficients are [1, 4, 14, 37, 92, 205, 433, 859, 1635, 2983, 5268, 9013, 15023, 24423, 38863, 60599, 92814, 139773, 207311, 303106, 437401, 623451, 878572, 1224839, 1690578, 2311417, 3132368, 4209365, 5612117, 7426306, 9757438, 12733909, 16512136, 21280753, 27267361, 34744174, 44036796, 55531702, 69687532, 87044878, 108240660] For the, 4, power which is the poset [{}, {1}, {1, 2}, {}, {4}, {1, 4, 5}, {}, {7}, {4, 7, 8}, {}, {10}, {7, 10, 11}, {}, {13}, {10, 13, 14}] the generating function for its P-partitions is -(1+4*q+4072198*q^53+4159318*q^38+2587121*q^33+3127027*q^56+2188302*q^59+ 5198059*q^48+5361526*q^46+5325804*q^44+5046147*q^49+4912836*q^41+1717062*q^30+ 3545369*q^36+3224657*q^35+3859094*q^37+1401*q^8+759*q^7+553160*q^24+38213*q^15+ 341825*q^22+4692523*q^40+2903334*q^34+5304090*q^47+5368801*q^45+4852334*q^50+ 199142*q^20+148406*q^19+5233626*q^43+4359360*q^52+17098*q^13+78246*q^17+55248*q ^16+4204*q^10+394*q^6+4439278*q^39+3449400*q^55+2491154*q^58+4621485*q^51+ 3766655*q^54+1901358*q^60+6908*q^11+14*q^2+36*q^3+88*q^4+2470*q^9+100*q^86+1734 *q^82+3077*q^81+5241*q^80+475*q^84+21194*q^77+13708*q^78+8614*q^79+191*q^5+q^90 +4*q^89+40*q^87+933*q^83+227*q^85+14*q^88+11024*q^12+108743*q^18+437902*q^23+ 262910*q^21+2805714*q^57+1029771*q^27+1990024*q^31+2281249*q^32+5094784*q^42+ 689311*q^25+1464951*q^29+25869*q^14+847864*q^26+1235557*q^28+790771*q^65+ 1633623*q^61+400584*q^68+1164669*q^63+509564*q^67+965747*q^64+1387543*q^62+ 639101*q^66+31911*q^76+94887*q^73+310298*q^69+236671*q^70+67406*q^74+131003*q^ 72+177601*q^71+46901*q^75)/(q^14-1)/(q^15-1)/(q^12-1)/(q^11-1)/(q^13-1)/(q^9-1) /(q^8-1)/(q^10-1)/(q^4+q^3-q-1)/(q^6-1)/(q-1)^2/(q^7-1)/(q^5-1)/(q^4-1) The first, 41, coefficients are [1, 5, 20, 61, 169, 420, 978, 2136, 4449, 8866, 17038, 31665, 57172, 100513, 172572, 289873, 477335, 771679, 1226599, 1919190, 2959271, 4500974, 6758963, 10028527, 14712911, 21357011, 30692127, 43690825, 61638364, 86219728, 119631009, 164714288, 225127722, 305550245, 411936374, 551821026, 734694634, 972449273, 1279922100, 1675538063, 2182085665] For the, 5, power which is the poset [{}, {1}, {1, 2}, {}, {4}, {1, 4, 5}, {}, {7}, {4, 7, 8}, {}, {10}, {7, 10, 11}, {}, {13}, {10, 13, 14}, {}, {16}, {13, 16, 17}] the generating function for its P-partitions is (1+5*q+1699906491*q^53+231526182*q^38+82689675*q^33+20*q^133+2134156077*q^56+ 2542809375*q^59+1031254533*q^48+807993114*q^46+616597103*q^44+5*q^134+ 1153896255*q^49+9914*q^126+470*q^130+1104*q^129+65*q^132+184*q^131+390389613*q^ 41+5024*q^127+170304*q^121+18741*q^125+40106072*q^30+157235199*q^36+128003689*q ^35+191562401*q^37+3988*q^8+1957*q^7+7130262*q^24+211493*q^15+3640307*q^22+ 330456619*q^40+677096*q^118+103330106*q^34+436835*q^119+915785830*q^47+ 708206913*q^45+1283030921*q^50+62204706*q^104+1756391*q^20+78711965*q^103+ 1192004*q^19+533184410*q^43+223534116*q^98+445698419*q^94+98682686*q^102+ 122618226*q^101+184501036*q^99+151044062*q^100+692800231*q^91+519929168*q^93+ 602253342*q^92+320450122*q^96+1557158771*q^52+28945723*q^107+78523*q^13+521606* q^17+335557*q^16+14535*q^10+915*q^6+277656053*q^39+1989966127*q^55+2412352304*q ^58+1417797588*q^51+1844676774*q^54+2665140379*q^60+26281*q^11+102519*q^122+ 34115*q^124+20*q^2+60*q^3+164*q^4+8956574*q^111+2252341*q^115+3251061*q^114+ 7759*q^9+1029044*q^117+1535523*q^116+1263375449*q^86+1824079447*q^82+1969612325 *q^81+2114257685*q^80+48690768*q^105+16483327*q^109+275910*q^120+1536675635*q^ 84+2525577474*q^77+2394015958*q^78+2256313192*q^79+6477887*q^112+4622074*q^113+ 37736035*q^106+400*q^5+791567491*q^90+898405639*q^89+1134880762*q^87+1679269948 *q^83+1397647061*q^85+1013004023*q^88+21965816*q^108+12225224*q^110+46082*q^12+ 795585*q^18+5128974*q^23+2547470*q^21+379318191*q^95+268679629*q^97+2275539243* q^57+17784397*q^27+51531956*q^31+65580990*q^32+457857691*q^42+9786579*q^25+ 30904767*q^29+130419*q^14+13269893*q^26+23570139*q^28+q^135+3098804398*q^65+ 2777628157*q^61+3173421449*q^68+2966704810*q^63+3165810221*q^67+3040468060*q^64 +2878648140*q^62+3140805535*q^66+2649220283*q^76+2955837166*q^73+3163515958*q^ 69+3136247846*q^70+2865926793*q^74+3031595674*q^72+3092044396*q^71+2763216613*q ^75+60041*q^123+2424*q^128)/(q^18-1)/(q^17-1)/(q^15-1)/(q^16-1)/(q^14-1)/(q^12-\ 1)/(q^13-1)/(q^11-1)/(q^9-1)/(q^4-1)/(q^8-1)/(q^10-1)/(q^4+q^3-q-1)/(q^6-1)/(q-\ 1)^2/(q^7-1)/(q^5-1) The first, 41, coefficients are [1, 6, 27, 93, 284, 776, 1969, 4674, 10532, 22654, 46852, 93534, 181019, 340581, 624718, 1119502, 1963879, 3377919, 5705330, 9474473, 15487413, 24945356, 39627094, 62136025, 96243985, 147359621, 223168115, 334489855, 496433923, 729925532, 1063727812, 1537085955, 2203181102, 3133596063, 4424079298, 6201919360, 8635360382, 11945531585, 16421528112, 22439347230, 30485609602] The whole thing took, 38.750, seconds of CPU time The Two-rowed, atomic object with, 3, cells in each row and the second row is jutting by, -1, cells compared to the one above, which in our notation is the poset [{}, {1}, {1, 2}, {}, {1, 4}, {1, 2, 4, 5}] The generating functions for up to, 6, rows i.e. up to the, 5, -th power of it are as follows This is the story of the P-partitions for the n-th power of the poset [{}, {1}, {1, 2}, {}, {1, 4}, {1, 2, 4, 5}] glued by the first and last, 3, vetrices for n from 1 to, 5 as well as the first , 41, terms of its counting series For the, 1, power which is the poset [{}, {1}, {1, 2}, {}, {1, 4}, {1, 2, 4, 5}] the generating function for its P-partitions is (q^6+q^5+q^4+q^3+q^2+q+1)/(q^4+q^3-q-1)/(q^5-1)/(q^4-1)/(q^3-1)/(q-1)^2 The first, 41, coefficients are [1, 2, 4, 8, 14, 23, 37, 55, 81, 116, 161, 219, 294, 386, 501, 642, 812, 1016, 1261, 1548, 1887, 2283, 2741, 3270, 3880, 4574, 5366, 6264, 7277, 8417, 9698, 11126, 12720, 14492, 16454, 18623, 21018, 23648, 26538, 29704, 33163] For the, 2, power which is the poset [{}, {1}, {1, 2}, {}, {1, 4}, {1, 2, 4, 5}, {}, {4, 7}, {1, 4, 5, 7, 8}] the generating function for its P-partitions is -(q^14+q^13+q^12+2*q^11+2*q^10+3*q^9+4*q^8+3*q^7+4*q^6+3*q^5+2*q^4+2*q^3+q^2+q+ 1)/(q^3-1)^2/(q^3-q^2-q+1)/(q^4-1)/(q^5-1)/(q^2-1)/(q^7-1)/(q^5-q^4+q^3-q^2+q-1 ) The first, 41, coefficients are [1, 3, 7, 16, 32, 60, 109, 187, 310, 499, 779, 1187, 1772, 2591, 3723, 5266, 7337, 10086, 13698, 18387, 24423, 32125, 41865, 54093, 69338, 88205, 111416, 139801, 174309, 216044, 266270, 326411, 398110, 483220, 583823, 702283, 841255, 1003690, 1192919, 1412642, 1666961] For the, 3, power which is the poset [{}, {1}, {1, 2}, {}, {1, 4}, {1, 2, 4, 5}, {}, {4, 7}, {1, 4, 5, 7, 8}, {}, {7, 10}, {4, 7, 8, 10, 11}] the generating function for its P-partitions is (1+2*q+q^30+36*q^8+27*q^7+20*q^24+87*q^15+36*q^22+57*q^20+66*q^19+81*q^13+81*q^ 17+85*q^16+57*q^10+20*q^6+66*q^11+3*q^2+6*q^3+9*q^4+47*q^9+13*q^5+75*q^12+75*q^ 18+27*q^23+47*q^21+6*q^27+13*q^25+2*q^29+85*q^14+9*q^26+3*q^28)/(q^8-q^5-q^3+1) /(q^5-q^4-q+1)/(q^8+q^6-q^2-1)/(q^6-q^5-q+1)/(q^3-q^2-q+1)/(q-1)^2/(q^2-1)/(q^4 +1)/(q^6+q^5+q^4+q^3+q^2+q+1)/(q^8+q^7+q^6+q^5+q^4+q^3+q^2+q+1)/(q^2+q+1) The first, 41, coefficients are [1, 4, 11, 28, 63, 131, 260, 490, 888, 1559, 2654, 4400, 7128, 11300, 17570, 26844, 40347, 59739, 87238, 125756, 179115, 252272, 351582, 485170, 663332, 898995, 1208342, 1611479, 2133215, 2804056, 3661288, 4750221, 6125740, 7853984, 10014278, 12701462, 16028439, 20129060, 25161564, 31312308, 38799998] For the, 4, power which is the poset [{}, {1}, {1, 2}, {}, {1, 4}, {1, 2, 4, 5}, {}, {4, 7}, {1, 4, 5, 7, 8}, {}, {7, 10}, {4, 7, 8, 10, 11}, {}, {10, 13}, {7, 10, 11, 13, 14}] the generating function for its P-partitions is -(1+2*q+24*q^38+122*q^33+q^44+6*q^41+228*q^30+50*q^36+69*q^35+34*q^37+50*q^8+34 *q^7+435*q^24+269*q^15+453*q^22+9*q^40+93*q^34+435*q^20+412*q^19+2*q^43+190*q^ 13+348*q^17+309*q^16+93*q^10+24*q^6+15*q^39+122*q^11+3*q^2+6*q^3+9*q^4+69*q^9+ 15*q^5+154*q^12+384*q^18+449*q^23+449*q^21+348*q^27+190*q^31+154*q^32+3*q^42+ 412*q^25+269*q^29+228*q^14+384*q^26+309*q^28)/(q^7+1-q^6-q)/(q^8+q^7-q^5-2*q^4- q^3+q+1)/(q^5-q^4-q+1)/(q^3-q^2-q+1)/(q^8+1-q^7-q)/(q^6-q^5-q+1)^2/(q^5-2*q^4+2 *q^3-2*q^2+2*q-1)/(q^4+1)/(q^10+q^9+q^8+q^7+q^6+q^5+q^4+q^3+q^2+q+1)/(q^8+q^7+q ^6+q^5+q^4+q^3+q^2+q+1)/(q^2+q+1)^2 The first, 41, coefficients are [1, 5, 16, 45, 112, 255, 548, 1117, 2180, 4107, 7494, 13296, 23014, 38943, 64557, 105034, 167955, 264305, 409806, 626665, 945959, 1410737, 2080020, 3034069, 4381104, 6265820, 8880311, 12477756, 17389505, 24046524, 33005868, 44983256, 60893251, 81898204, 109467712, 145450885, 192163353, 252491737, 330019061, 429174144, 555409151] For the, 5, power which is the poset [{}, {1}, {1, 2}, {}, {1, 4}, {1, 2, 4, 5}, {}, {4, 7}, {1, 4, 5, 7, 8}, {}, {7, 10}, {4, 7, 8, 10, 11}, {}, {10, 13}, {7, 10, 11, 13, 14}, {}, {13, 16}, {10, 13, 14, 16, 17}] the generating function for its P-partitions is (1+4*q+37243*q^53+189836*q^38+173110*q^33+18279*q^56+7659*q^59+89675*q^48+ 115504*q^46+141111*q^44+77414*q^49+173110*q^41+141111*q^30+189836*q^36+186373*q ^35+191001*q^37+466*q^8+280*q^7+65902*q^24+7659*q^15+45725*q^22+180731*q^40+ 180731*q^34+102462*q^47+128501*q^45+65902*q^50+29861*q^20+23559*q^19+152977*q^ 43+45725*q^52+3880*q^13+13934*q^17+10430*q^16+1179*q^10+162*q^6+186373*q^39+ 23559*q^55+10430*q^58+55299*q^51+29861*q^54+5509*q^60+1797*q^11+10*q^2+23*q^3+ 47*q^4+753*q^9+89*q^5+2673*q^12+18279*q^18+55299*q^23+37243*q^21+13934*q^57+ 102462*q^27+152977*q^31+163757*q^32+163757*q^42+77414*q^25+128501*q^29+5509*q^ 14+89675*q^26+115504*q^28+753*q^65+3880*q^61+162*q^68+1797*q^63+280*q^67+1179*q ^64+2673*q^62+466*q^66+4*q^73+89*q^69+47*q^70+q^74+10*q^72+23*q^71)/(q^20+1-q^ 11-q^9)/(q^6-1)/(q^2-2*q+1)/(q^3-q^2-q+1)/(q^6+1-q^4-q^2)/(q^12+1-q^7-q^5)/(q^ 18+1-q^10-q^8)/(q-1)/(q^6-q^5-q+1)/(q^8+1-q^7-q)/(q^2+1)/(q^4+q^2+1)/(q^2+q+1)^ 3/(q^12+q^11+q^10+q^9+q^8+q^7+q^6+q^5+q^4+q^3+q^2+q+1)/(q^4-q^2+1) The first, 41, coefficients are [1, 6, 22, 68, 185, 457, 1056, 2307, 4809, 9644, 18694, 35169, 64440, 115290, 201849, 346490, 584059, 968097, 1579803, 2540745, 4030873, 6313674, 9770905, 14950382, 22631003, 33910447, 50322926, 73995889, 107857090, 155906905, 223573873, 318176125, 449517469, 630652769, 878865619, 1216911785, 1674592481, 2290735912, 3115683063, 4214392000, 5670298914] The whole thing took, 37.383, seconds of CPU time The Two-rowed, atomic object with, 3, cells in each row and the second row is jutting by, 0, cells compared to the one above, which in our notation is the poset [{}, {1}, {1, 2}, {1}, {1, 2, 4}, {1, 2, 3, 4, 5}] The generating functions for up to, 6, rows i.e. up to the, 5, -th power of it are as follows This is the story of the P-partitions for the n-th power of the poset [{}, {1}, {1, 2}, {1}, {1, 2, 4}, {1, 2, 3, 4, 5}] glued by the first and last, 3, vetrices for n from 1 to, 5 as well as the first , 41, terms of its counting series For the, 1, power which is the poset [{}, {1}, {1, 2}, {1}, {1, 2, 4}, {1, 2, 3, 4, 5}] the generating function for its P-partitions is 1/(q^4-1)/(q^4+q^3-q-1)/(q-1)^2/(q^3-1)/(q^2-1) The first, 41, coefficients are [1, 1, 3, 5, 9, 13, 22, 30, 45, 61, 85, 111, 150, 190, 247, 309, 390, 478, 593, 715, 870, 1038, 1243, 1465, 1735, 2023, 2368, 2740, 3175, 3643, 4189, 4771, 5443, 6163, 6982, 7858, 8852, 9908, 11098, 12366, 13780] For the, 2, power which is the poset [{}, {1}, {1, 2}, {1}, {1, 2, 4}, {1, 2, 3, 4, 5}, {1, 4}, {1, 2, 4, 5, 7}, {1, 2, 3, 4, 5, 6, 7, 8}] the generating function for its P-partitions is -1/(q^3-q^2-q+1)/(q^5-1)/(q^3-1)^2/(q^4-1)^2/(q-1)/(q^4+q^3-q-1) The first, 41, coefficients are [1, 1, 3, 6, 11, 18, 32, 49, 78, 117, 174, 250, 360, 499, 692, 940, 1266, 1678, 2215, 2878, 3723, 4763, 6057, 7633, 9575, 11906, 14742, 18129, 22191, 27001, 32728, 39446, 47373, 56623, 67437, 79976, 94548, 111327, 130701, 152909, 178380] For the, 3, power which is the poset [{}, {1}, {1, 2}, {1}, {1, 2, 4}, {1, 2, 3, 4, 5}, {1, 4}, {1, 2, 4, 5, 7}, {1, 2, 3, 4, 5, 6, 7, 8}, {1, 4, 7}, {1, 2, 4, 5, 7, 8, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}] the generating function for its P-partitions is 1/(q^5-q^4-q+1)/(q^6+1-q^4-q^2)/(q^5-q^3-q^2+1)/(q-1)^3/(q^2-1)^2/(q^3-q^2+q-1) /(q^4+q^3+q+q^2+1)^2/(q^4+q^2+1)/(q^2+q+1)^2 The first, 41, coefficients are [1, 1, 3, 6, 12, 20, 37, 59, 99, 154, 241, 361, 545, 790, 1148, 1628, 2298, 3182, 4392, 5962, 8061, 10761, 14292, 18785, 24578, 31857, 41109, 52640, 67100, 84954, 107116, 134236, 167566, 208054, 257349, 316799, 388621, 474626, 577761, 700519, 846700] For the, 4, power which is the poset [{}, {1}, {1, 2}, {1}, {1, 2, 4}, {1, 2, 3, 4, 5}, {1, 4}, {1, 2, 4, 5, 7}, {1, 2, 3, 4, 5, 6, 7, 8}, {1, 4, 7}, {1, 2, 4, 5, 7, 8, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 4, 7, 10}, {1, 2, 4, 5, 7, 8, 10, 11, 13}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}] the generating function for its P-partitions is -1/(q^5-q^4-q+1)/(q^8-q^6-q^2+1)/(q^7+1-q^6-q)/(q^6+1-q^4-q^2)/(q^7+q^6-2*q^4-2 *q^3+q+1)/(q^2-2*q+1)^2/(q-1)/(q^4+q^3+q+q^2+1)^3/(q^2+q+1)/(q^2+1)/(q^6+q^5+q^ 4+q^3+q^2+q+1) The first, 41, coefficients are [1, 1, 3, 6, 12, 21, 39, 64, 109, 175, 280, 432, 667, 997, 1486, 2171, 3146, 4489, 6365, 8901, 12370, 17006, 23223, 31427, 42276, 56407, 74841, 98611, 129230, 168312, 218138, 281121, 360634, 460308, 584992, 740042, 932447, 1169896, 1462335, 1820777, 2259112] For the, 5, power which is the poset [{}, {1}, {1, 2}, {1}, {1, 2, 4}, {1, 2, 3, 4, 5}, {1, 4}, {1, 2, 4, 5, 7}, {1, 2, 3, 4, 5, 6, 7, 8}, {1, 4, 7}, {1, 2, 4, 5, 7, 8, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 4, 7, 10}, {1, 2, 4, 5, 7, 8, 10, 11, 13}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, {1, 4, 7, 10, 13}, {1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}] the generating function for its P-partitions is 1/(q^8-q^6-q^2+1)/(q^11-q^8-q^3+1)/(q^6+1-q^4-q^2)^2/(q^5-q^4-q+1)/(q^9-q^6-q^3 +1)/(q^2-2*q+1)^3/(q^4+q^3+q+q^2+1)^3/(q^6+q^5+q^4+q^3+q^2+q+1)^2/(q^2+q+1)^2/( q^2-q+1) The first, 41, coefficients are [1, 1, 3, 6, 12, 21, 40, 66, 114, 185, 301, 471, 740, 1123, 1704, 2531, 3736, 5425, 7838, 11162, 15810, 22145, 30830, 42522, 58324, 79332, 107338, 144201, 192723, 255952, 338307, 444601, 581674, 757109, 981260, 1265837, 1626463, 2080816, 2652164, 3367005, 4259446] The whole thing took, 13.841, seconds of CPU time The Two-rowed, atomic object with, 3, cells in each row and the second row is jutting by, 1, cells compared to the one above, which in our notation is the poset [{}, {1}, {1, 2}, {1, 2}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}] The generating functions for up to, 6, rows i.e. up to the, 5, -th power of it are as follows This is the story of the P-partitions for the n-th power of the poset [{}, {1}, {1, 2}, {1, 2}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}] glued by the first and last, 3, vetrices for n from 1 to, 5 as well as the first , 41, terms of its counting series For the, 1, power which is the poset [{}, {1}, {1, 2}, {1, 2}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}] the generating function for its P-partitions is 1/(q^4+q^3-q-1)/(q^5-1)/(q^4-1)/(q^3-1)/(q-1)^2 The first, 41, coefficients are [1, 1, 2, 4, 6, 9, 14, 19, 27, 37, 49, 64, 84, 106, 134, 168, 207, 253, 309, 371, 445, 530, 626, 736, 863, 1003, 1163, 1343, 1543, 1766, 2017, 2291, 2597, 2935, 3305, 3712, 4161, 4647, 5181, 5763, 6394] For the, 2, power which is the poset [{}, {1}, {1, 2}, {1, 2}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5, 6, 7, 8}] the generating function for its P-partitions is -(q^4-q^2+1)/(q^4-1)/(q^9-1)/(q^6-q^4+q^2-1)/(q^7-1)/(q^4+q^3-q-1)/(q^5-1)/(q^3 -1)/(q-1)^2 The first, 41, coefficients are [1, 1, 2, 4, 6, 9, 15, 21, 31, 45, 62, 85, 117, 155, 206, 271, 351, 451, 578, 729, 917, 1146, 1420, 1750, 2149, 2617, 3176, 3837, 4610, 5516, 6579, 7806, 9234, 10885, 12782, 14962, 17466, 20315, 23569, 27269, 31459] For the, 3, power which is the poset [{}, {1}, {1, 2}, {1, 2}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5, 6, 7, 8}, {1, 2, 3, 4, 5, 6, 7, 8}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}] the generating function for its P-partitions is (q^6-q^3+1)/(q^4-1)/(q^7-1)/(q^6+q^5+q^4-q^2-q-1)/(q^11-1)/(q^10-1)/(q^9-1)/(q^ 6-q^4+q^2-1)/(q^3-1)^2/(q^5-1)/(q-1)^2 The first, 41, coefficients are [1, 1, 2, 4, 6, 9, 15, 21, 31, 46, 64, 89, 125, 168, 227, 305, 402, 527, 690, 889, 1143, 1461, 1851, 2334, 2933, 3656, 4542, 5619, 6913, 8472, 10349, 12578, 15240, 18403, 22136, 26543, 31738, 37814, 44934, 53248, 62914] For the, 4, power which is the poset [{}, {1}, {1, 2}, {1, 2}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5, 6, 7, 8}, {1, 2, 3, 4, 5, 6, 7, 8}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}] the generating function for its P-partitions is -(q^8-q^4+1)*(q^6-q^3+1)/(q^2-1)/(q^5-1)/(q^4-1)/(q^15-1)/(q^14-1)/(q^13-1)/(q^ 6+q^5+q^4-q^2-q-1)/(q^11-1)/(q^10-1)/(q^7-1)/(q^9-1)/(q-1)^2/(q^3-1)^2 The first, 41, coefficients are [1, 1, 2, 4, 6, 9, 15, 21, 31, 46, 64, 89, 126, 170, 231, 313, 415, 548, 724, 940, 1219, 1574, 2013, 2564, 3257, 4103, 5154, 6450, 8027, 9954, 12308, 15142, 18576, 22717, 27675, 33616, 40724, 49163, 59201, 71102, 85149] For the, 5, power which is the poset [{}, {1}, {1, 2}, {1, 2}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5, 6, 7, 8}, {1, 2, 3, 4, 5, 6, 7, 8}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}] the generating function for its P-partitions is (q^8+q^7-q^5-q^4-q^3+q+1)*(q^8-q^4+1)/(q^4-1)/(q^7-1)/(q^6+q^5-q-1)/(q^13-1)/(q ^10+q^9-q-1)/(q^17-1)/(q^16-1)/(q^15-1)/(q^14-1)/(q^6+q^5+q^4-q^2-q-1)/(q^11-1) /(q^9-1)/(q-1)^3/(q^3-1)^2/(q^5-1) The first, 41, coefficients are [1, 1, 2, 4, 6, 9, 15, 21, 31, 46, 64, 89, 126, 170, 231, 314, 417, 552, 732, 953, 1240, 1608, 2064, 2640, 3370, 4265, 5384, 6775, 8476, 10570, 13148, 16272, 20086, 24723, 30314, 37068, 45216, 54965, 66658, 80641, 97281] The whole thing took, 4.828, seconds of CPU time The Two-rowed, atomic object with, 3, cells in each row and the second row is jutting by, 2, cells compared to the one above, which in our notation is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}] The generating functions for up to, 6, rows i.e. up to the, 5, -th power of it are as follows This is the story of the P-partitions for the n-th power of the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}] glued by the first and last, 3, vetrices for n from 1 to, 5 as well as the first , 41, terms of its counting series For the, 1, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}] the generating function for its P-partitions is 1/(q-1)/(q^2-1)/(q^3-1)/(q^4-1)/(q^5-1)/(q^6-1) The first, 41, coefficients are [1, 1, 2, 3, 5, 7, 11, 14, 20, 26, 35, 44, 58, 71, 90, 110, 136, 163, 199, 235, 282, 331, 391, 454, 532, 612, 709, 811, 931, 1057, 1206, 1360, 1540, 1729, 1945, 2172, 2432, 2702, 3009, 3331, 3692] For the, 2, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5, 6, 7, 8}] the generating function for its P-partitions is -1/(q-1)/(q^2-1)/(q^3-1)/(q^4-1)/(q^5-1)/(q^6-1)/(q^7-1)/(q^8-1)/(q^9-1) The first, 41, coefficients are [1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 54, 73, 94, 123, 157, 201, 252, 318, 393, 488, 598, 732, 887, 1076, 1291, 1549, 1845, 2194, 2592, 3060, 3589, 4206, 4904, 5708, 6615, 7657, 8824, 10156, 11648, 13338] For the, 3, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5, 6, 7, 8}, {1, 2, 3, 4, 5, 6, 7, 8, 9}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}] the generating function for its P-partitions is 1/(q-1)/(q^2-1)/(q^3-1)/(q^4-1)/(q^5-1)/(q^6-1)/(q^7-1)/(q^8-1)/(q^9-1)/(q^10-1 )/(q^11-1)/(q^12-1) The first, 41, coefficients are [1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 100, 133, 172, 224, 285, 366, 460, 582, 725, 905, 1116, 1380, 1686, 2063, 2503, 3036, 3655, 4401, 5262, 6290, 7476, 8877, 10489, 12384, 14552, 17084, 19978, 23334] For the, 4, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5, 6, 7, 8}, {1, 2, 3, 4, 5, 6, 7, 8, 9}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}] the generating function for its P-partitions is -1/(q-1)/(q^2-1)/(q^3-1)/(q^4-1)/(q^5-1)/(q^6-1)/(q^7-1)/(q^8-1)/(q^9-1)/(q^10-\ 1)/(q^11-1)/(q^12-1)/(q^13-1)/(q^14-1)/(q^15-1) The first, 41, coefficients are [1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 230, 295, 381, 483, 615, 773, 972, 1210, 1508, 1861, 2297, 2815, 3446, 4192, 5096, 6158, 7434, 8932, 10715, 12801, 15272, 18148, 21535, 25469, 30073] For the, 5, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5, 6, 7, 8}, {1, 2, 3, 4, 5, 6, 7, 8, 9}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}] the generating function for its P-partitions is 1/(q-1)/(q^2-1)/(q^3-1)/(q^4-1)/(q^5-1)/(q^6-1)/(q^7-1)/(q^8-1)/(q^9-1)/(q^10-1 )/(q^11-1)/(q^12-1)/(q^13-1)/(q^14-1)/(q^15-1)/(q^16-1)/(q^17-1)/(q^18-1) The first, 41, coefficients are [1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 489, 625, 788, 995, 1243, 1556, 1928, 2391, 2943, 3621, 4426, 5409, 6570, 7976, 9635, 11626, 13968, 16765, 20040, 23928, 28472, 33834] The whole thing took, 3.420, seconds of CPU time The Two-rowed, atomic object with, 4, cells in each row and the second row is jutting by, -3, cells compared to the one above, which in our notation is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {}, {5}, {5, 6}, {1, 5, 6, 7}] The generating functions for up to, 6, rows i.e. up to the, 5, -th power of it are as follows This is the story of the P-partitions for the n-th power of the poset [{}, {1}, {1, 2}, {1, 2, 3}, {}, {5}, {5, 6}, {1, 5, 6, 7}] glued by the first and last, 4, vetrices for n from 1 to, 5 as well as the first , 41, terms of its counting series For the, 1, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {}, {5}, {5, 6}, {1, 5, 6, 7}] the generating function for its P-partitions is (1+2*q^2+4*q^4+q+q^16+7*q^10+3*q^3+5*q^11+7*q^7+5*q^12+2*q^14+3*q^13+5*q^5+7*q^ 9+q^15+8*q^8+7*q^6)/(q-1)/(q^2-1)/(q^3-1)/(q^4-1)/(q^5-1)/(q^6-1)/(q^7-1)/(q^8-\ 1) The first, 41, coefficients are [1, 2, 5, 10, 19, 33, 57, 91, 144, 217, 323, 465, 663, 921, 1267, 1710, 2287, 3011, 3934, 5074, 6499, 8236, 10371, 12942, 16060, 19773, 24221, 29470, 35690, 42966, 51509, 61425, 72969, 86276, 101650, 119259, 139467, 162476, 188717, 218438, 252142] For the, 2, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {}, {5}, {5, 6}, {1, 5, 6, 7}, {}, {9}, {9, 10}, {5, 9, 10, 11}] the generating function for its P-partitions is (1+2*q+262*q^38+879*q^33+q^48+5*q^46+20*q^44+86*q^41+1373*q^30+461*q^36+584*q^ 35+350*q^37+121*q^8+80*q^7+1914*q^24+839*q^15+1832*q^22+131*q^40+729*q^34+2*q^ 47+10*q^45+1623*q^20+1480*q^19+33*q^43+552*q^13+1163*q^17+1002*q^16+244*q^10+52 *q^6+186*q^39+328*q^11+5*q^2+10*q^3+18*q^4+174*q^9+31*q^5+434*q^12+1329*q^18+ 1886*q^23+1737*q^21+1765*q^27+1207*q^31+1046*q^32+56*q^42+1896*q^25+1520*q^29+ 691*q^14+1852*q^26+1659*q^28)/(q^3-q^2-q+1)/(q^12-1)/(q^11-1)/(q^10-1)/(q^6-1)/ (q^9-1)/(q^8-1)/(q^7-1)/(q^5-1)/(q-1)/(q^4-1)/(q^2+q+1) The first, 41, coefficients are [1, 3, 9, 22, 49, 101, 199, 370, 665, 1151, 1937, 3168, 5069, 7927, 12172, 18351, 27240, 39822, 57450, 81811, 115173, 160339, 220983, 301610, 408019, 547244, 728203, 961630, 1260923, 1642098, 2124901, 2732752, 3494216, 4442957, 5619605, 7071764, 8856379, 11039767, 13700628, 16930135, 20835658] For the, 3, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {}, {5}, {5, 6}, {1, 5, 6, 7}, {}, {9}, {9, 10}, {5, 9, 10, 11}, {}, {13}, {13, 14}, {9, 13, 14, 15}] the generating function for its P-partitions is (1+3*q+2010016*q^53+1445404*q^38+850555*q^33+1722870*q^56+1368960*q^59+2206902* q^48+2164877*q^46+2056007*q^44+2201334*q^49+1789701*q^41+553755*q^30+1200875*q^ 36+1080168*q^35+1323190*q^37+611*q^8+347*q^7+177490*q^24+13488*q^15+110697*q^22 +1680832*q^40+962958*q^34+2194562*q^47+2118253*q^45+2178263*q^50+65494*q^20+ 49306*q^19+1979263*q^43+9*q^94+104*q^91+22*q^93+51*q^92+q^96+2081522*q^52+6307* q^13+26683*q^17+19149*q^16+1695*q^10+190*q^6+1565265*q^39+1828780*q^55+1490775* q^58+2137929*q^51+1925257*q^54+1246296*q^60+2694*q^11+9*q^2+22*q^3+48*q^4+1034* q^9+1870*q^86+10286*q^82+14853*q^81+21058*q^80+4613*q^84+53847*q^77+40044*q^78+ 29277*q^79+98*q^5+205*q^90+377*q^89+1135*q^87+6961*q^83+2973*q^85+670*q^88+4176 *q^12+36572*q^18+141043*q^23+85725*q^21+3*q^95+1609383*q^57+329319*q^27+645129* q^31+744339*q^32+1889945*q^42+220621*q^25+470527*q^29+9323*q^14+271091*q^26+ 395757*q^28+680036*q^65+1124508*q^61+421085*q^68+891005*q^63+499078*q^67+782188 *q^64+1005629*q^62+585549*q^66+71350*q^76+152348*q^73+351462*q^69+290219*q^70+ 119926*q^74+191164*q^72+236894*q^71+93124*q^75)/(q^15-1)/(q^16-1)/(q^12-1)/(q^ 11-1)/(q^13-1)/(q^14-1)/(q^8-1)/(q^9-1)/(q^10-1)/(q-1)^3/(q^7-1)/(q^4+q^3-q-1)/ (q^6-1)/(q^5-1)/(q^3+q^2+q+1) The first, 41, coefficients are [1, 4, 14, 40, 102, 239, 527, 1098, 2192, 4208, 7816, 14089, 24745, 42426, 71202, 117149, 189309, 300832, 470761, 726139, 1105215, 1661220, 2467883, 3625982, 5272605, 7592105, 10831267, 15317186, 21481450, 29888643, 41274142, 56588485, 77055333, 104239255, 140132924, 187258870, 248798600, 328742713, 432079276, 565012057, 735231377] For the, 4, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {}, {5}, {5, 6}, {1, 5, 6, 7}, {}, {9}, {9, 10}, {5, 9, 10, 11}, {}, {13}, {13, 14}, {9, 13, 14, 15}, {}, {17}, {17, 18}, {13, 17, 18, 19}] the generating function for its P-partitions is (1+4*q+1531510790*q^53+134521184*q^38+43435548*q^33+9587682*q^133+2151669627*q^ 56+2894233965*q^59+784813432*q^48+578437838*q^46+416730524*q^44+7090033*q^134+ 906630742*q^49+61048882*q^126+22352669*q^130+29099842*q^129+12835227*q^132+ 17016899*q^131+243650082*q^41+48077077*q^127+180235104*q^121+76920903*q^125+ 20062106*q^30+87498465*q^36+69808428*q^35+108875514*q^37+2084*q^8+1057*q^7+ 3330503*q^24+97944*q^15+1678396*q^22+201201824*q^40+317937013*q^118+55278806*q^ 34+264824201*q^119+675652313*q^47+492399209*q^45+1041739381*q^50+2261898145*q^ 104+803915*q^20+2503347002*q^103+544880*q^19+350602378*q^43+3874433182*q^98+ 5064500156*q^94+2757408586*q^102+3022992670*q^101+3583188346*q^99+3298776983*q^ 100+1897694*q^138+5913077502*q^91+5356011074*q^93+5639795562*q^92+4469139565*q^ 96+1353839002*q^52+1620488842*q^107+37081*q^13+239055*q^17+154428*q^16+7217*q^ 10+513*q^6+165068438*q^39+1930605676*q^55+2634328846*q^58+1190670686*q^51+ 1723793598*q^54+3164998767*q^60+12793*q^11+147217464*q^122+96194689*q^124+14*q^ 2+40*q^3+101*q^4+967798023*q^111+530351327*q^115+621619458*q^114+3946*q^9+ 379339637*q^117+449866725*q^116+7028068395*q^86+7481727608*q^82+7520862397*q^81 +7528825424*q^80+2033901232*q^105+1265387677*q^109+219187141*q^120+7312150353*q ^84+7366587807*q^77+7451223301*q^78+7505516400*q^79+839630334*q^112+724462631*q ^113+1819971011*q^106+235*q^5+6173101380*q^90+6417148876*q^89+6847012662*q^87+ 7411878485*q^83+7183711021*q^85+6642612697*q^88+617890*q^141+1435634996*q^108+ 1109558702*q^110+68904*q^146+22061*q^12+363818*q^18+2378564*q^23+1169375*q^21+ 4767988820*q^95+4170487517*q^97+2386484962*q^57+8552140*q^27+26174508*q^31+ 33857901*q^32+293183581*q^42+4610136*q^25+15240841*q^29+60913*q^14+6312034*q^26 +11471806*q^28+175133*q^144+412754*q^142+41838*q^147+5187811*q^135+271203*q^143 +910997*q^140+4623046308*q^65+3445172478*q^61+5502585730*q^68+4026798708*q^63+ 5214874071*q^67+4324225604*q^64+3733076649*q^62+4920797475*q^66+7252621808*q^76 +6749654049*q^73+5781173753*q^69+6047875559*q^70+6942361539*q^74+6534722680*q^ 72+6299945229*q^71+7110659747*q^75+119423620*q^123+37559493*q^128+24814*q^148+ 14323*q^149+110956*q^145+1145*q^153+8038*q^150+4360*q^151+2285*q^152+549*q^154+ 40*q^157+q^160+14*q^158+4*q^159+105*q^156+247*q^155+1323766*q^139+3754329*q^136 +2685468*q^137)/(q^20-1)/(q^19-1)/(q^18-1)/(q^17-1)/(q^16-1)/(q^15-1)/(q^14-1)/ (q^13-1)/(q^12-1)/(q^11-1)/(q^4-1)/(q^8-1)/(q^3-1)/(q^9-1)/(q^6+q^5-q-1)/(q^10-\ 1)/(q-1)^2/(q^7-1)/(q^6-1) The first, 41, coefficients are [1, 5, 20, 65, 186, 485, 1179, 2700, 5893, 12333, 24893, 48654, 92424, 171092, 309416, 547732, 950780, 1620755, 2716851, 4483532, 7291852, 11698074, 18527422, 28990952, 44849268, 68637413, 103974597, 155983634, 231859719, 341630861, 499174830, 723562616, 1040832830, 1486310183, 2107632814, 2968668196, 4154573032, 5778277119, 7988781459, 10981699308, 15012627378] For the, 5, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {}, {5}, {5, 6}, {1, 5, 6, 7}, {}, {9}, {9, 10}, {5, 9, 10, 11}, {}, {13}, {13, 14}, {9, 13, 14, 15}, {}, {17}, {17, 18}, {13, 17, 18, 19}, {}, {21}, {21, 22}, {17, 21, 22, 23}] the generating function for its P-partitions is (16800646*q^218+10837731*q^219+6892779*q^220+4318107*q^221+25697412*q^217+ 2662146*q^222+1613256*q^223+959865*q^224+51304*q^229+6219*q^232+13039*q^231+ 178330*q^227+559877*q^225+500*q^235+1229*q^234+26320*q^230+38810802*q^216+ 57915192*q^215+180093266*q^212+65*q^237+20*q^238+5*q^239+q^240+190*q^236+1+5*q+ 138987364413*q^53+3830403612*q^38+851248376*q^33+49338751142702*q^133+ 249184288180*q^56+430059040164*q^59+47936981493*q^48+30250786917*q^46+ 18688046886*q^44+47774832973302*q^134+59884939685*q^49+57705822942545*q^126+ 53546168318617*q^130+54757358434803*q^129+50828110862831*q^132+52233576053565*q ^131+8696758708*q^41+56844198873350*q^127+59987285690319*q^121+58438144615764*q ^125+314390676*q^30+2144967285*q^36+1588154807*q^35+2876317445*q^37+5700*q^8+ 2634*q^7+33272818*q^24+486228*q^15+14402881*q^22+6659110552*q^40+59638390645157 *q^118+1167196076*q^34+59899466619411*q^119+38179379893*q^47+23841608152*q^45+ 74442770416*q^50+43425986661014*q^104+5916495*q^20+41674633915418*q^103+3711107 *q^19+14566550084*q^43+32661538627350*q^98+25662949892336*q^94+39893668455455*q ^102+38092687424824*q^101+34467780056116*q^99+36281045899634*q^100+ 40967491844682*q^138+20834521657347*q^91+24004075028300*q^93+22393050103242*q^ 92+29102418141615*q^96+113397817771*q^52+48404505368579*q^107+156381*q^13+ 1391998*q^17+830588*q^16+23604*q^10+1159*q^6+5066879125*q^39+206012155871*q^55+ 360000443063*q^58+92094904535*q^51+169586357873*q^54+511705543099*q^60+45616*q^ 11+59813403164374*q^122+59036121416187*q^124+20*q^2+65*q^3+185*q^4+ 54038833323795*q^111+58011411953664*q^115+57200407528777*q^114+11817*q^9+ 59234645999482*q^117+58691058431666*q^116+13946431457711*q^86+9620896057230*q^ 82+8705339798043*q^81+7853879453630*q^80+45137968299790*q^105+51396340348597*q^ 109+60016039481416*q^120+11649575090411*q^84+5665196324883*q^77+6335966346854*q ^78+7064767109733*q^79+55207385033029*q^112+56263609481825*q^113+46800746685805 *q^106+480*q^5+19332480729161*q^90+17890265433473*q^89+15195471016457*q^87+ 10601962421943*q^83+12764348079901*q^85+16510571955724*q^88+35555473581362*q^ 141+49939537675074*q^108+52765709470903*q^110+26674743062620*q^146+85598*q^12+ 2291817*q^18+22027248*q^23+9295193*q^21+27364376485868*q^95+30870519907525*q^97 +300131813581*q^57+107094061*q^27+441934261*q^31+615893496*q^32+11288738240*q^ 42+49671238*q^25+221659923*q^29+278844*q^14+73325523*q^26+154826563*q^28+ 30157571219814*q^144+33743171487400*q^142+24988788174676*q^147+46145954450442*q ^135+31941199154373*q^143+37369618829203*q^140+1152236835907*q^65+606471841279* q^61+1795067191768*q^68+842183000498*q^63+1553903772180*q^67+986888968739*q^64+ 716027673073*q^62+1340455046372*q^66+5049975541747*q^76+3510646795240*q^73+ 2066542782306*q^69+2371028023719*q^70+3975500769157*q^74+3090214768031*q^72+ 2711303758322*q^71+4487663458200*q^75+59495607890212*q^123+55859157096622*q^128 +43019063396*q^193+126569093926*q^188+16537702367*q^197+12840384027*q^198+ 9910569089*q^199+7602545327*q^200+21176987733*q^196+5795301363*q^201+4388964188 *q^202+3301589491*q^203+1829223788*q^205+712748786*q^208+257547341*q^211+ 364841856*q^210+983625816*q^207+1346575315*q^206+23347607539869*q^148+ 2466381320*q^204+85442924*q^214+124691432*q^213+512165670*q^209+21756249489778* q^149+28399777942607*q^145+15967115258139*q^153+20219107901170*q^150+ 18739919846018*q^151+17321772477526*q^152+14677780217842*q^154+11211322802611*q ^157+8345599338536*q^160+10190221816433*q^158+9235378817628*q^159+ 12299470930514*q^156+13455006409440*q^155+6049544782931*q^163+7519327287305*q^ 161+5401519181068*q^164+4808057953626*q^165+4266462644065*q^166+6754690448049*q ^162+3773930776001*q^167+3327593052727*q^168+2924546955053*q^169+2236738364974* q^171+1458447445814*q^174+921195936699*q^177+1077557506350*q^176+1687727761941* q^173+1946269965242*q^172+2561888206522*q^170+1255870642850*q^175+784608661595* q^178+473816698472*q^181+665757400545*q^179+397356212197*q^182+331885049460*q^ 183+276057066949*q^184+562745254583*q^180+228653203850*q^185+188575417536*q^186 +154839733001*q^187+102985722570*q^189+53911159002*q^192+26965881918*q^195+ 34150161082*q^194+67220546167*q^191+83403562291*q^190+39176706954325*q^139+ 44461881828588*q^136+42732454100910*q^137+319690*q^226+2834*q^233+97004*q^228)/ (q^24-1)/(q^21-1)/(q^20-1)/(q^23-1)/(q^22-1)/(q^16-1)/(q^18-1)/(q^17-1)/(q^19-1 )/(q^15-1)/(q^14-1)/(q^13-1)/(q^12-1)/(q^3-1)/(q^9-1)/(q^4-1)/(q^8-1)/(q^6+q^5- q-1)/(q^10-1)/(q-1)^3/(q^11-1)/(q^7-1)/(q^5+q^4+q^3+q^2+q+1) The first, 41, coefficients are [1, 6, 27, 98, 310, 887, 2350, 5843, 13789, 31112, 67520, 141588, 287977, 569813, 1099706, 2074537, 3832447, 6944399, 12359710, 21633686, 37280217, 63310405, 106047929, 175348796, 286410520, 462427861, 738460790, 1167016161, 1826044105, 2830299628, 4347377144, 6620171942, 9998148757, 14980576615, 22275963442, 32883256034, 48202177610, 70182307996, 101523496656, 145943872248, 208536561908] The whole thing took, 608.026, seconds of CPU time The Two-rowed, atomic object with, 4, cells in each row and the second row is jutting by, -2, cells compared to the one above, which in our notation is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {}, {5}, {1, 5, 6}, {1, 2, 5, 6, 7}] The generating functions for up to, 6, rows i.e. up to the, 5, -th power of it are as follows This is the story of the P-partitions for the n-th power of the poset [{}, {1}, {1, 2}, {1, 2, 3}, {}, {5}, {1, 5, 6}, {1, 2, 5, 6, 7}] glued by the first and last, 4, vetrices for n from 1 to, 5 as well as the first , 41, terms of its counting series For the, 1, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {}, {5}, {1, 5, 6}, {1, 2, 5, 6, 7}] the generating function for its P-partitions is (q^12+q^11+2*q^10+3*q^9+4*q^8+4*q^7+4*q^6+3*q^5+3*q^4+2*q^3+2*q^2+q+1)/(q^4-1)^ 2/(q^7-1)/(q^6-1)/(q^5-1)/(q^3-1)/(q^2-1)/(q-1) The first, 41, coefficients are [1, 2, 5, 9, 18, 30, 52, 82, 130, 194, 289, 414, 591, 818, 1126, 1516, 2029, 2667, 3486, 4492, 5756, 7289, 9182, 11453, 14217, 17499, 21442, 26084, 31599, 38037, 45612, 54391, 64630, 76416, 90055, 105659, 123591, 143989, 167280, 193639, 223563] For the, 2, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {}, {5}, {1, 5, 6}, {1, 2, 5, 6, 7}, {}, {9}, {5, 9, 10}, {5, 6, 9, 10, 11}] the generating function for its P-partitions is (1+q+q^30+25*q^8+17*q^7+18*q^24+58*q^15+31*q^22+45*q^20+49*q^19+52*q^13+56*q^17 +61*q^16+38*q^10+14*q^6+42*q^11+3*q^2+3*q^3+7*q^4+29*q^9+8*q^5+51*q^12+56*q^18+ 23*q^23+36*q^21+5*q^27+12*q^25+q^29+59*q^14+9*q^26+3*q^28)/(q^8+q^6-q^2-1)/(q-1 )^3/(q^7-1)/(q^5-1)/(q^10-1)/(q^9-1)/(q^8-1)/(q^4+q^3-q-1)/(q^2-1)/(q^4-1) The first, 41, coefficients are [1, 3, 9, 20, 45, 89, 173, 314, 559, 952, 1590, 2573, 4092, 6352, 9709, 14559, 21537, 31362, 45125, 64077, 90028, 125067, 172111, 234538, 316924, 424572, 564491, 744802, 976011, 1270271, 1643028, 2112111, 2699827, 3431857, 4339902, 5460354, 6837599, 8522435, 10576168, 13068748, 16083791] For the, 3, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {}, {5}, {1, 5, 6}, {1, 2, 5, 6, 7}, {}, {9}, {5, 9, 10}, {5, 6, 9, 10, 11}, {}, {13}, {9, 13, 14}, {9, 10, 13, 14, 15}] the generating function for its P-partitions is (1+3*q+1371*q^53+28229*q^38+34222*q^33+399*q^56+79*q^59+6108*q^48+9518*q^46+ 13779*q^44+4746*q^49+21145*q^41+32507*q^30+31808*q^36+33048*q^35+30175*q^37+273 *q^8+169*q^7+20068*q^24+3408*q^15+15196*q^22+23639*q^40+33870*q^34+7696*q^47+ 11551*q^45+3611*q^50+10809*q^20+8890*q^19+16151*q^43+1945*q^52+1872*q^13+5713*q ^17+4459*q^16+640*q^10+102*q^6+26020*q^39+623*q^55+143*q^58+2682*q^51+940*q^54+ 41*q^60+937*q^11+8*q^2+16*q^3+32*q^4+423*q^9+58*q^5+1343*q^12+7194*q^18+17592*q ^23+12915*q^21+244*q^57+27230*q^27+33526*q^31+34113*q^32+18631*q^42+22540*q^25+ 31073*q^29+2556*q^14+24959*q^26+29300*q^28+19*q^61+3*q^63+q^64+8*q^62)/(q^8-1)/ (q^7+q^6+q^5-q^2-q-1)/(q^3-q^2-q+1)^2/(q^8+q^7-q-1)/(q^13-1)/(q^12-1)/(q^11-1)/ (q^9-1)/(q^2-1)/(q^6-1)/(q^10-1)/(q^3-q^2+q-1)/(q^7-1)/(q^2+1) The first, 41, coefficients are [1, 4, 14, 37, 93, 209, 450, 910, 1780, 3341, 6102, 10813, 18734, 31700, 52625, 85695, 137279, 216382, 336245, 515300, 779900, 1166174, 1724577, 2523296, 3655656, 5246091, 7461956, 10523614, 14722907, 20439662, 28170072, 38552984, 52412776, 70800237, 95056356, 126874571, 168394293, 222293919, 291926119, 381454475, 496046699] For the, 4, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {}, {5}, {1, 5, 6}, {1, 2, 5, 6, 7}, {}, {9}, {5, 9, 10}, {5, 6, 9, 10, 11}, {}, {13}, {9, 13, 14}, {9, 10, 13, 14, 15}, {}, {17}, {13, 17, 18}, {13, 14, 17, 18, 19}] the generating function for its P-partitions is (1+2*q+408040*q^53+304022*q^38+182756*q^33+348477*q^56+275899*q^59+451829*q^48+ 445046*q^46+424640*q^44+449739*q^49+372577*q^41+121107*q^30+254581*q^36+229887* q^35+279268*q^37+208*q^8+121*q^7+40730*q^24+3535*q^15+25962*q^22+351115*q^40+ 205986*q^34+450082*q^47+436327*q^45+444358*q^50+15764*q^20+12013*q^19+409724*q^ 43+q^94+14*q^91+2*q^93+6*q^92+423273*q^52+1736*q^13+6720*q^17+4938*q^16+522*q^ 10+74*q^6+327953*q^39+370263*q^55+300848*q^58+435312*q^51+390394*q^54+250983*q^ 60+788*q^11+6*q^2+10*q^3+23*q^4+327*q^9+332*q^86+1946*q^82+2831*q^81+4053*q^80+ 850*q^84+10523*q^77+7798*q^78+5665*q^79+39*q^5+31*q^90+59*q^89+195*q^87+1299*q^ 83+536*q^85+111*q^88+1195*q^12+9080*q^18+32659*q^23+20316*q^21+325061*q^57+ 73509*q^27+140144*q^31+160870*q^32+392418*q^42+50085*q^25+103509*q^29+2517*q^14 +61068*q^26+87745*q^28+136217*q^65+226166*q^61+84090*q^68+178845*q^63+99748*q^ 67+156892*q^64+202113*q^62+117206*q^66+14012*q^76+30154*q^73+70064*q^69+57797*q ^70+23694*q^74+37940*q^72+47078*q^71+18335*q^75)/(q^13-1)/(q^14-1)/(q^3-1)/(q^9 -1)/(q^5-1)^3/(q^11-q^10+q^6-q^5+q-1)/(q^16-1)/(q^8-q^7+q^6-q^5+q^3-q^2+q-1)/(q -1)^2/(q^6-1)/(q^12-1)/(q^2-1)^3/(q^11-1)/(q^4-1)/(q^8-1)/(q^6+q^5+q^4+q^3+q^2+ q+1) The first, 41, coefficients are [1, 5, 20, 61, 170, 425, 1002, 2219, 4712, 9594, 18914, 36142, 67303, 122285, 217494, 379093, 648939, 1092060, 1809374, 2954014, 4757546, 7563940, 11881739, 18451915, 28348386, 43108744, 64922299, 96874020, 143286925, 210164412, 305798354, 441550688, 632908891, 900834895, 1273558624, 1788856473, 2497039652, 3464727399, 4779735957, 6557214037, 8947506358] For the, 5, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {}, {5}, {1, 5, 6}, {1, 2, 5, 6, 7}, {}, {9}, {5, 9, 10}, {5, 6, 9, 10, 11}, {}, {13}, {9, 13, 14}, {9, 10, 13, 14, 15}, {}, {17}, {13, 17, 18}, {13, 14, 17, 18, 19}, {}, {21}, {17, 21, 22}, {17, 18, 21, 22, 23}] the generating function for its P-partitions is (1+4*q+2051081073*q^53+163149121*q^38+50230092*q^33+4389548*q^133+2914400864*q^ 56+3953255150*q^59+1024846001*q^48+746096135*q^46+530268351*q^44+3081332*q^134+ 1190675160*q^49+37740640*q^126+11784209*q^130+16008908*q^129+6174052*q^132+ 8579061*q^131+303082018*q^41+28637454*q^127+131256721*q^121+49274550*q^125+ 22487129*q^30+104197010*q^36+82341766*q^35+130861105*q^37+2010*q^8+1023*q^7+ 3496400*q^24+94567*q^15+1724490*q^22+248247661*q^40+251715107*q^118+64569569*q^ 34+204126317*q^119+877013271*q^47+630918510*q^45+1375484110*q^50+2384127836*q^ 104+809230*q^20+2679303448*q^103+543140*q^19+442898461*q^43+4439732587*q^98+ 6073925482*q^94+2995016826*q^102+3330367668*q^101+4054562451*q^99+3684100616*q^ 100+648487*q^138+7302872513*q^91+6489565189*q^93+6900577031*q^92+5244177714*q^ 96+1805151950*q^52+1625581590*q^107+35455*q^13+234141*q^17+150150*q^16+6893*q^ 10+505*q^6+201951800*q^39+2605974313*q^55+3589353266*q^58+1580091837*q^51+ 2318062625*q^54+4332405908*q^60+12186*q^11+104029992*q^122+63757571*q^124+15*q^ 2+40*q^3+103*q^4+900560609*q^111+452337651*q^115+542518168*q^114+3772*q^9+ 308165987*q^117+374637473*q^116+9038143930*q^86+9867186812*q^82+9972068094*q^81 +10032537188*q^80+2110022958*q^105+1224250227*q^109+164312050*q^120+9530055650* q^84+9942666970*q^77+10017722223*q^78+10047809957*q^79+765381476*q^112+ 646443808*q^113+1857206274*q^106+232*q^5+7692290095*q^90+8064613492*q^89+ 8741484783*q^87+9719216313*q^83+9302077809*q^85+8415696752*q^88+168661*q^141+ 1414795795*q^108+1053158061*q^110+11277*q^146+21056*q^12+359429*q^18+2469962*q^ 23+1188923*q^21+5657560705*q^95+4837192585*q^97+3242564960*q^57+9278621*q^27+ 29650425*q^31+38756363*q^32+367576352*q^42+4892878*q^25+16900255*q^29+58521*q^ 14+6773516*q^26+12583657*q^28+36119*q^144+103367*q^142+5989*q^147+2133980*q^135 +61876*q^143+269409*q^140+6365443579*q^65+4724537026*q^61+7573201870*q^68+ 5536927994*q^63+7180638600*q^67+5950925775*q^64+5127039464*q^62+6776684854*q^66 +9823652236*q^76+9220900118*q^73+7950182991*q^69+8307436367*q^70+9460446113*q^ 74+8946590259*q^72+8640870897*q^71+9662212955*q^75+81783527*q^123+21520254*q^ 128+3062*q^148+1494*q^149+20492*q^145+45*q^153+694*q^150+302*q^151+123*q^152+15 *q^154+q^156+4*q^155+421826*q^139+1457049*q^136+979866*q^137)/(q^16+q^12-q^4-1) /(q-1)^3/(q^12+q^11-q-1)/(q^9+q^8+q^7-q^2-q-1)/(q^12+q^10-q^2-1)/(q^19-1)/(q^18 -1)/(q^17-1)/(q^3-q^2-q+1)^4/(q^16-1)/(q^15-1)/(q^14-1)/(q^13-1)/(q^7-1)/(q^3+2 *q^2+2*q+1)/(q^6-1)/(q^4+q^3+q+q^2+1)^2/(q^2+1)^3/(q^4-q^3+q^2-q+1)/(q^6+q^3+1) The first, 41, coefficients are [1, 6, 27, 93, 285, 782, 2001, 4799, 10968, 23985, 50597, 103276, 204917, 396140, 748301, 1383602, 2509072, 4468515, 7826736, 13496604, 22938328, 38455809, 63648792, 104075648, 168241251, 269024709, 425764855, 667233641, 1035902991, 1593950163, 2431740486, 3679621422, 5524347017, 8231652546, 12177289473, 17889222464, 26104961340, 37848724213, 54535147829, 78107543176, 111221866321] The whole thing took, 923.582, seconds of CPU time The Two-rowed, atomic object with, 4, cells in each row and the second row is jutting by, -1, cells compared to the one above, which in our notation is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {}, {1, 5}, {1, 2, 5, 6}, {1, 2, 3, 5, 6, 7}] The generating functions for up to, 6, rows i.e. up to the, 5, -th power of it are as follows This is the story of the P-partitions for the n-th power of the poset [{}, {1}, {1, 2}, {1, 2, 3}, {}, {1, 5}, {1, 2, 5, 6}, {1, 2, 3, 5, 6, 7}] glued by the first and last, 4, vetrices for n from 1 to, 5 as well as the first , 41, terms of its counting series For the, 1, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {}, {1, 5}, {1, 2, 5, 6}, {1, 2, 3, 5, 6, 7}] the generating function for its P-partitions is (q^6+q^3+1)/(q^4-1)^2/(q-1)^2/(q^6-1)/(q^5-1)/(q^3-1)/(q^2-1) The first, 41, coefficients are [1, 2, 4, 8, 15, 25, 42, 66, 102, 152, 223, 318, 449, 619, 844, 1133, 1506, 1973, 2565, 3296, 4204, 5311, 6665, 8295, 10265, 12610, 15409, 18713, 22617, 27182, 32530, 38737, 45947, 54258, 63842, 74818, 87395, 101713, 118018, 136486, 157402] For the, 2, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {}, {1, 5}, {1, 2, 5, 6}, {1, 2, 3, 5, 6, 7}, {}, {5, 9}, {1, 5, 6, 9, 10}, {1, 2, 5, 6, 7, 9, 10, 11}] the generating function for its P-partitions is (1+2*q+18*q^8+15*q^7+q^24+21*q^15+3*q^22+7*q^20+9*q^19+26*q^13+15*q^17+18*q^16+ 24*q^10+12*q^6+26*q^11+3*q^2+5*q^3+7*q^4+21*q^9+9*q^5+27*q^12+12*q^18+2*q^23+5* q^21+24*q^14)/(q^8+q^6-q^2-1)/(q^5-1)/(q^6-1)/(q^3-1)^2/(q-1)^2/(q^6+q^5-q-1)/( q^8-1)/(q^7-1)/(q^4-1)/(q^2-1) The first, 41, coefficients are [1, 3, 7, 16, 33, 63, 117, 207, 354, 589, 955, 1512, 2349, 3579, 5363, 7915, 11518, 16539, 23469, 32923, 45704, 62823, 85560, 115508, 154674, 205507, 271055, 355026, 461953, 597304, 767724, 981135, 1247074, 1576858, 1983964, 2484280, 3096614, 3842956, 4749166, 5845343, 7166603] For the, 3, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {}, {1, 5}, {1, 2, 5, 6}, {1, 2, 3, 5, 6, 7}, {}, {5, 9}, {1, 5, 6, 9, 10}, {1, 2, 5, 6, 7, 9, 10, 11}, {}, {9, 13}, {5, 9, 10, 13, 14}, {1, 5, 6, 9, 10, 11, 13, 14, 15}] the generating function for its P-partitions is (1+3*q+226*q^38+766*q^33+q^48+6*q^46+21*q^44+81*q^41+1212*q^30+395*q^36+504*q^ 35+302*q^37+117*q^8+81*q^7+1721*q^24+766*q^15+1656*q^22+117*q^40+628*q^34+3*q^ 47+12*q^45+1475*q^20+1351*q^19+34*q^43+504*q^13+1063*q^17+913*q^16+226*q^10+54* q^6+165*q^39+302*q^11+6*q^2+12*q^3+21*q^4+165*q^9+34*q^5+395*q^12+1212*q^18+ 1704*q^23+1579*q^21+1579*q^27+1063*q^31+913*q^32+54*q^42+1704*q^25+1351*q^29+ 628*q^14+1656*q^26+1475*q^28)/(q^5-q^3-q^2+1)/(q^9-q^5-q^4+1)/(q^11-q^6-q^5+1)/ (q-1)^3/(q^5-1)/(q^6-q^5+q-1)/(q^9-1)/(q^8-1)^2/(q^3-1)/(q^8+q^7-q-1)/(q^7+q^6+ 2*q^5+2*q^4+2*q^3+2*q^2+q+1)/(q^6+q^5+q^4+q^3+q^2+q+1) The first, 41, coefficients are [1, 4, 11, 28, 64, 135, 272, 523, 968, 1738, 3035, 5171, 8624, 14102, 22650, 35793, 55717, 85526, 129598, 194016, 287178, 420579, 609803, 875827, 1246701, 1759618, 2463617, 3422934, 4721184, 6466589, 8798492, 11895271, 15984203, 21353437, 28366585, 37480429, 49266412, 64436308, 83873307, 108669013, 140167636] For the, 4, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {}, {1, 5}, {1, 2, 5, 6}, {1, 2, 3, 5, 6, 7}, {}, {5, 9}, {1, 5, 6, 9, 10}, {1, 2, 5, 6, 7, 9, 10, 11}, {}, {9, 13}, {5, 9, 10, 13, 14}, {1, 5, 6, 9, 10, 11, 13, 14, 15}, {}, {13, 17}, {9, 13, 14, 17, 18}, {5, 9, 10, 13, 14, 15, 17, 18, 19}] the generating function for its P-partitions is (1+3*q+31173*q^53+94824*q^38+77901*q^33+18024*q^56+9218*q^59+60394*q^48+72386*q ^46+82872*q^44+54196*q^49+93243*q^41+60394*q^30+90670*q^36+87170*q^35+93243*q^ 37+220*q^8+139*q^7+26314*q^24+3061*q^15+18024*q^22+94824*q^40+82872*q^34+66502* q^47+77901*q^45+48071*q^50+11687*q^20+9218*q^19+87170*q^43+36459*q^52+1599*q^13 +5476*q^17+4129*q^16+524*q^10+84*q^6+95363*q^39+21918*q^55+11687*q^58+42127*q^ 51+26314*q^54+7162*q^60+773*q^11+6*q^2+14*q^3+27*q^4+346*q^9+3*q^77+q^78+47*q^5 +1125*q^12+7162*q^18+21918*q^23+14619*q^21+14619*q^57+42127*q^27+66502*q^31+ 72386*q^32+90670*q^42+31173*q^25+54196*q^29+2228*q^14+36459*q^26+48071*q^28+ 1599*q^65+5476*q^61+524*q^68+3061*q^63+773*q^67+2228*q^64+4129*q^62+1125*q^66+6 *q^76+47*q^73+346*q^69+220*q^70+27*q^74+84*q^72+139*q^71+14*q^75)/(q^12+q^10-q^ 2-1)/(q-1)^4/(q^11-1)/(q^12+q^9-q^3-1)/(q^7-1)^2/(q^10-1)/(q^3-1)/(q^9-1)/(q^4-\ 1)^3/(q^8-1)/(q^7+q^6+q^5-q^2-q-1)/(q^2-1)/(q^10-q^8+q^6-q^4+q^2-1)/(q^6-q^4+q^ 2-1)/(q^4+q^2+1)/(q^4+q^3+q+q^2+1) The first, 41, coefficients are [1, 5, 16, 45, 113, 260, 565, 1168, 2315, 4434, 8241, 14915, 26372, 45660, 77560, 129483, 212748, 344440, 550084, 867380, 1351471, 2082287, 3174622, 4791971, 7165373, 10618780, 15603157, 22741983, 32891481, 47220076, 67312623, 95306204, 134066462, 187415365, 260423850, 359786429, 494298609, 675462473, 918252227, 1242078009, 1671994507] For the, 5, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {}, {1, 5}, {1, 2, 5, 6}, {1, 2, 3, 5, 6, 7}, {}, {5, 9}, {1, 5, 6, 9, 10}, {1, 2, 5, 6, 7, 9, 10, 11}, {}, {9, 13}, {5, 9, 10, 13, 14}, {1, 5, 6, 9, 10, 11, 13, 14, 15}, {}, {13, 17}, {9, 13, 14, 17, 18}, {5, 9, 10, 13, 14, 15, 17, 18, 19}, {}, {17, 21}, {13, 17, 18, 21, 22}, {9, 13, 14, 17, 18, 19, 21, 22, 23}] the generating function for its P-partitions is (1+5*q+83501257*q^53+17863437*q^38+7252467*q^33+93598880*q^56+98571326*q^59+ 59982959*q^48+49979430*q^46+40441697*q^44+65003838*q^49+27828369*q^41+3806213*q ^30+12771528*q^36+10666178*q^35+15166164*q^37+1276*q^8+707*q^7+809781*q^24+ 36633*q^15+444396*q^22+24194861*q^40+15*q^118+8833077*q^34+5*q^119+54952405*q^ 47+45124786*q^45+69942658*q^50+54605*q^104+232752*q^20+80137*q^103+165333*q^19+ 35975155*q^43+444396*q^98+1412524*q^94+115904*q^102+165333*q^101+323563*q^99+ 232752*q^100+3014282*q^91+1836711*q^93+2364470*q^92+809781*q^96+79268788*q^52+ 15667*q^107+15667*q^13+80137*q^17+54605*q^16+3763*q^10+375*q^6+20872011*q^39+ 90733174*q^55+97556760*q^58+74723507*q^51+87346123*q^54+98911787*q^60+6194*q^11 +15*q^2+39*q^3+90*q^4+2225*q^111+189*q^115+375*q^114+2225*q^9+39*q^117+90*q^116 +8833077*q^86+17863437*q^82+20872011*q^81+24194861*q^80+36633*q^105+6194*q^109+ q^120+12771528*q^84+35975155*q^77+31761504*q^78+27828369*q^79+1276*q^112+707*q^ 113+24172*q^106+189*q^5+3806213*q^90+4761629*q^89+7252467*q^87+15166164*q^83+ 10666178*q^85+5902813*q^88+9957*q^108+3763*q^110+9957*q^12+115904*q^18+603274*q ^23+323563*q^21+1075181*q^95+603274*q^97+95888296*q^57+1836711*q^27+4761629*q^ 31+5902813*q^32+31761504*q^42+1075181*q^25+3014282*q^29+24172*q^14+1412524*q^26 +2364470*q^28+90733174*q^65+98571326*q^61+79268788*q^68+95888296*q^63+83501257* q^67+93598880*q^64+97556760*q^62+87346123*q^66+40441697*q^76+54952405*q^73+ 74723507*q^69+69942658*q^70+49979430*q^74+59982959*q^72+65003838*q^71+45124786* q^75)/(q-1)^7/(q^14-1)/(q^12+q^9-q^3-1)/(q^8-1)^2/(q^10-1)^2/(q^4-1)/(q^12-1)/( q^8+q^7+q^6+q^5-q^3-q^2-q-1)/(q^4+q^3-q-1)/(q^6-1)^2/(q^9-q^6+q^3-1)/(q^11-1)/( q^9+q^8+q^7-q^2-q-1)/(q^7+q^6+q^5-q^2-q-1)/(q^13-1)/(q^10+q^9+q^8+q^7+q^6+q^5+q ^4+q^3+q^2+q+1)/(q^6+q^3+1)/(q^6+q^5+q^4+q^3+q^2+q+1) The first, 41, coefficients are [1, 6, 22, 68, 186, 463, 1079, 2382, 5024, 10204, 20059, 38316, 71364, 129944, 231821, 405967, 698958, 1184713, 1979190, 3262230, 5309826, 8541397, 13588310, 21392556, 33347800, 51499331, 78825752, 119633219, 180103418, 269050860, 398963343, 587423040, 859037275, 1248048094, 1801840995, 2585640102, 3688761425, 5232902696, 7383085289, 10362035074, 14469004169] The whole thing took, 455.824, seconds of CPU time The Two-rowed, atomic object with, 4, cells in each row and the second row is jutting by, 0, cells compared to the one above, which in our notation is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1}, {1, 2, 5}, {1, 2, 3, 5, 6}, {1, 2, 3, 4, 5, 6, 7}] The generating functions for up to, 6, rows i.e. up to the, 5, -th power of it are as follows This is the story of the P-partitions for the n-th power of the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1}, {1, 2, 5}, {1, 2, 3, 5, 6}, {1, 2, 3, 4, 5, 6, 7}] glued by the first and last, 4, vetrices for n from 1 to, 5 as well as the first , 41, terms of its counting series For the, 1, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1}, {1, 2, 5}, {1, 2, 3, 5, 6}, {1, 2, 3, 4, 5, 6, 7}] the generating function for its P-partitions is 1/(q^5-1)/(q^4+q^3-q-1)/(q^4-1)^2/(q-1)^2/(q^3-1)/(q^2-1) The first, 41, coefficients are [1, 1, 3, 5, 10, 15, 26, 38, 60, 85, 125, 172, 243, 325, 442, 580, 767, 986, 1275, 1612, 2045, 2548, 3179, 3910, 4812, 5849, 7109, 8554, 10285, 12259, 14599, 17255, 20372, 23895, 27991, 32603, 37925, 43890, 50725, 58361, 67053 ] For the, 2, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1}, {1, 2, 5}, {1, 2, 3, 5, 6}, {1, 2, 3, 4, 5, 6, 7}, {1, 5}, {1, 2, 5, 6, 9}, {1, 2, 3, 5, 6, 7, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}] the generating function for its P-partitions is 1/(q^4-1)^2/(q-1)^4/(q^8+q^6-q^2-1)/(q^6+q^5-q-1)/(q^3-1)^2/(q^4+q^3-q-1)/(q^5-\ 1)/(q+1) The first, 41, coefficients are [1, 1, 3, 6, 12, 20, 37, 59, 99, 154, 241, 361, 545, 790, 1148, 1628, 2298, 3182, 4392, 5962, 8061, 10761, 14292, 18785, 24578, 31857, 41109, 52640, 67100, 84954, 107116, 134236, 167566, 208054, 257349, 316799, 388621, 474626, 577761, 700519, 846700] For the, 3, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1}, {1, 2, 5}, {1, 2, 3, 5, 6}, {1, 2, 3, 4, 5, 6, 7}, {1, 5}, {1, 2, 5, 6, 9}, {1, 2, 3, 5, 6, 7, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 5, 9}, {1, 2, 5, 6, 9, 10, 13}, {1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}] the generating function for its P-partitions is 1/(q^7-1)/(q^6-1)/(q^5-1)^2/(q^2-1)^2/(q-1)^4/(q^4-1)^2/(q^7+q^6+q^5-q^2-q-1)/( q^8+q^6-q^2-1)/(q^3-1)^2/(q^3+q^2+q+1)/(q+1) The first, 41, coefficients are [1, 1, 3, 6, 13, 22, 42, 70, 122, 197, 322, 502, 789, 1194, 1808, 2673, 3935, 5683, 8173, 11574, 16305, 22689, 31396, 43001, 58581, 79096, 106237, 141612, 187811, 247408, 324375, 422733, 548445, 707716, 909367, 1162775, 1480892, 1877612, 2371702, 2983552, 3740004] For the, 4, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1}, {1, 2, 5}, {1, 2, 3, 5, 6}, {1, 2, 3, 4, 5, 6, 7}, {1, 5}, {1, 2, 5, 6, 9}, {1, 2, 3, 5, 6, 7, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 5, 9}, {1, 2, 5, 6, 9, 10, 13}, {1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, {1, 5, 9, 13}, {1, 2, 5, 6, 9, 10, 13, 14, 17}, {1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}] the generating function for its P-partitions is 1/(q-1)^5/(q^7-1)^2/(q^6-1)^3/(q^8-1)/(q^6+q^5+q^4-q^2-q-1)/(q^4-1)^2/(q^8+q^7+ q^6+q^5-q^3-q^2-q-1)/(q^7+q^6+q^5-q^2-q-1)/(q^2-1)/(q^6+q^5-q-1)/(q^3-1)/(q^5-1 ) The first, 41, coefficients are [1, 1, 3, 6, 13, 23, 44, 75, 133, 220, 367, 587, 942, 1463, 2268, 3441, 5190, 7697, 11349, 16504, 23854, 34090, 48413, 68107, 95241, 132086, 182144, 249372, 339567, 459425, 618447, 827704, 1102506, 1460860, 1927046, 2529844, 3307301, 4304615, 5580645, 7205448, 9268887] For the, 5, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1}, {1, 2, 5}, {1, 2, 3, 5, 6}, {1, 2, 3, 4, 5, 6, 7}, {1, 5}, {1, 2, 5, 6, 9}, {1, 2, 3, 5, 6, 7, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 5, 9}, {1, 2, 5, 6, 9, 10, 13}, {1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, {1, 5, 9, 13}, {1, 2, 5, 6, 9, 10, 13, 14, 17}, {1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}, {1, 5, 9, 13, 17}, {1, 2, 5, 6, 9, 10, 13, 14, 17, 18, 21}, {1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23}] the generating function for its P-partitions is 1/(q^3-1)/(q^9-1)/(q^6+q^5-q-1)/(q-1)^7/(q^8+q^6-q^2-1)/(q^7-1)^2/(q^8+q^7+q^6+ q^5-q^3-q^2-q-1)/(q^2-1)^2/(q^9+q^8+q^7-q^2-q-1)/(q^12+q^10+q^8-q^4-q^2-1)/(q^6 -1)/(q^8-1)/(q^7+q^6+q^5-q^2-q-1)/(q^5-1)/(q^4-1)^2/(q^2+q+1)/(q^3+1)/(q+1) The first, 41, coefficients are [1, 1, 3, 6, 13, 23, 45, 77, 138, 231, 390, 632, 1029, 1620, 2549, 3926, 6014, 9061, 13579, 20072, 29497, 42874, 61933, 88643, 126133, 178025, 249864, 348232, 482725, 664968, 911424, 1242130, 1684846, 2273588, 3054395, 4083988, 5437836, 7208881, 9519174, 12519054, 16403260] The whole thing took, 108.403, seconds of CPU time The Two-rowed, atomic object with, 4, cells in each row and the second row is jutting by, 1, cells compared to the one above, which in our notation is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2}, {1, 2, 3, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}] The generating functions for up to, 6, rows i.e. up to the, 5, -th power of it are as follows This is the story of the P-partitions for the n-th power of the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2}, {1, 2, 3, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}] glued by the first and last, 4, vetrices for n from 1 to, 5 as well as the first , 41, terms of its counting series For the, 1, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2}, {1, 2, 3, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}] the generating function for its P-partitions is (q^3+q^4+q^8+q^5+1)/(q-1)/(q^2-1)/(q^3-1)/(q^4-1)/(q^5-1)/(q^6-1)/(q^7-1)/(q^8-\ 1) The first, 41, coefficients are [1, 1, 2, 4, 7, 11, 17, 25, 38, 53, 75, 103, 141, 187, 248, 323, 419, 534, 679, 852, 1066, 1318, 1624, 1984, 2416, 2917, 3512, 4200, 5008, 5935, 7014, 8246, 9670, 11285, 13137, 15231, 17617, 20298, 23337, 26740, 30576] For the, 2, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2}, {1, 2, 3, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 5, 6}, {1, 2, 3, 4, 5, 6, 7, 9}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}] the generating function for its P-partitions is (q^24+q^21+q^20+q^19+q^18+q^17+2*q^16+2*q^15+q^14+2*q^13+3*q^12+2*q^11+q^10+2*q ^9+2*q^8+q^7+q^6+q^5+q^4+q^3+1)/(q^5-1)/(q^4-1)/(q^12-1)/(q^11-1)/(q^10-1)/(q^9 -1)/(q^8-1)/(q^7-1)/(q^6-1)/(q^3-1)/(q-1)^2/(q+1) The first, 41, coefficients are [1, 1, 2, 4, 7, 11, 18, 27, 42, 62, 90, 129, 185, 256, 354, 484, 655, 875, 1164, 1530, 2002, 2596, 3345, 4281, 5454, 6895, 8680, 10867, 13541, 16786, 20729, 25473, 31190, 38029, 46199, 55914, 67456, 81080, 97166, 116070, 138248] For the, 3, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2}, {1, 2, 3, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 5, 6}, {1, 2, 3, 4, 5, 6, 7, 9}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 2, 3, 4, 5, 6, 7, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}] the generating function for its P-partitions is (1+2*q^38+5*q^33+q^48+q^44+q^41+5*q^30+4*q^36+4*q^35+3*q^37+2*q^8+q^7+9*q^24+5* q^15+6*q^22+2*q^40+3*q^34+q^45+7*q^20+6*q^19+q^43+4*q^13+5*q^17+6*q^16+2*q^10+q ^6+2*q^39+3*q^11+q^3+q^4+2*q^9+q^5+4*q^12+5*q^18+7*q^23+7*q^21+7*q^27+5*q^31+6* q^32+q^42+7*q^25+6*q^29+3*q^14+6*q^26+7*q^28)/(q^4-1)/(q^5-1)/(q^9-1)/(q^8-1)/( q^16-1)/(q^15-1)/(q^14-1)/(q^13-1)/(q^12-1)/(q^11-1)/(q^10-1)/(q^7-1)/(q^3-1)^2 /(q^4-q^3+q-1)/(q^2-1) The first, 41, coefficients are [1, 1, 2, 4, 7, 11, 18, 27, 42, 62, 91, 131, 189, 265, 370, 512, 703, 952, 1285, 1717, 2283, 3011, 3949, 5146, 6678, 8607, 11046, 14105, 17934, 22690, 28601, 35893, 44883, 55902, 69386, 85814, 105798, 129988, 159233, 194459, 236803] For the, 4, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2}, {1, 2, 3, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 5, 6}, {1, 2, 3, 4, 5, 6, 7, 9}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 2, 3, 4, 5, 6, 7, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}] the generating function for its P-partitions is (1+19*q^53+27*q^38+25*q^33+16*q^56+11*q^59+26*q^48+24*q^46+29*q^44+23*q^49+29*q ^41+21*q^30+29*q^36+27*q^35+28*q^37+2*q^8+q^7+16*q^24+6*q^15+11*q^22+31*q^40+24 *q^34+25*q^47+27*q^45+21*q^50+11*q^20+9*q^19+28*q^43+21*q^52+4*q^13+7*q^17+7*q^ 16+2*q^10+q^6+29*q^39+16*q^55+11*q^58+21*q^51+16*q^54+11*q^60+3*q^11+q^3+q^4+2* q^9+q^80+q^77+q^5+4*q^12+7*q^18+13*q^23+11*q^21+13*q^57+19*q^27+23*q^31+26*q^32 +27*q^42+16*q^25+21*q^29+4*q^14+16*q^26+21*q^28+6*q^65+9*q^61+4*q^68+7*q^63+4*q ^67+7*q^64+7*q^62+4*q^66+q^76+q^73+3*q^69+2*q^70+q^74+2*q^72+2*q^71+q^75)/(q^4-\ 1)/(q^7-1)/(q^8-1)/(q^11-1)/(q^12-1)/(q^15-1)/(q^20-1)/(q^19-1)/(q^18-1)/(q^17-\ 1)/(q^16-1)/(q^14-1)/(q^13-1)/(q^10-1)/(q^9-1)/(q^6-1)/(q^5-1)/(q-1)/(q^3-1)/(q ^2-1) The first, 41, coefficients are [1, 1, 2, 4, 7, 11, 18, 27, 42, 62, 91, 131, 189, 265, 371, 514, 707, 961, 1301, 1745, 2331, 3088, 4071, 5335, 6963, 9031, 11667, 15001, 19212, 24493, 31116, 39369, 49647, 62374, 78113, 97499, 121336, 150519, 186202, 229681, 282561] For the, 5, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2}, {1, 2, 3, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 5, 6}, {1, 2, 3, 4, 5, 6, 7, 9}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 2, 3, 4, 5, 6, 7, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21}, { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23}] the generating function for its P-partitions is (1+116*q^53+59*q^38+43*q^33+125*q^56+125*q^59+105*q^48+91*q^46+88*q^44+104*q^49 +73*q^41+32*q^30+56*q^36+50*q^35+57*q^37+2*q^8+q^7+20*q^24+6*q^15+13*q^22+72*q^ 40+44*q^34+97*q^47+91*q^45+105*q^50+7*q^104+12*q^20+7*q^103+10*q^19+82*q^43+13* q^98+21*q^94+8*q^102+10*q^101+13*q^99+12*q^100+30*q^91+25*q^93+28*q^92+20*q^96+ 116*q^52+4*q^107+4*q^13+7*q^17+7*q^16+2*q^10+q^6+66*q^39+120*q^55+121*q^58+112* q^51+116*q^54+129*q^60+3*q^11+q^3+q^4+2*q^111+q^115+q^114+2*q^9+q^117+q^116+44* q^86+59*q^82+66*q^81+72*q^80+6*q^105+3*q^109+q^120+56*q^84+82*q^77+76*q^78+73*q ^79+2*q^112+q^113+4*q^106+q^5+32*q^90+36*q^89+43*q^87+57*q^83+50*q^85+41*q^88+4 *q^108+2*q^110+4*q^12+8*q^18+16*q^23+13*q^21+20*q^95+16*q^97+124*q^57+25*q^27+ 36*q^31+41*q^32+76*q^42+20*q^25+30*q^29+4*q^14+21*q^26+28*q^28+120*q^65+125*q^ 61+116*q^68+124*q^63+116*q^67+125*q^64+121*q^62+116*q^66+88*q^76+97*q^73+112*q^ 69+105*q^70+91*q^74+105*q^72+104*q^71+91*q^75)/(q^4-1)/(q^5-1)/(q^8-1)/(q^9-1)/ (q^12-1)/(q^13-1)/(q^17-1)/(q^16-1)/(q^24-1)/(q^23-1)/(q^22-1)/(q^21-1)/(q^20-1 )/(q^19-1)/(q^18-1)/(q^15-1)/(q^11-1)/(q^14-1)/(q^7-1)/(q^10-1)/(q-1)/(q^3-1)^2 /(q^4-q^3+q-1)/(q+1) The first, 41, coefficients are [1, 1, 2, 4, 7, 11, 18, 27, 42, 62, 91, 131, 189, 265, 371, 514, 707, 961, 1302, 1747, 2335, 3097, 4087, 5363, 7011, 9108, 11789, 15190, 19497, 24917, 31738, 40267, 50929, 64186, 80645, 101006, 126155, 157084, 195082, 241613, 298486] The whole thing took, 39.194, seconds of CPU time The Two-rowed, atomic object with, 4, cells in each row and the second row is jutting by, 2, cells compared to the one above, which in our notation is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}] The generating functions for up to, 6, rows i.e. up to the, 5, -th power of it are as follows This is the story of the P-partitions for the n-th power of the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}] glued by the first and last, 4, vetrices for n from 1 to, 5 as well as the first , 41, terms of its counting series For the, 1, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}] the generating function for its P-partitions is 1/(q^4-1)^2/(q^7-1)/(q^6-1)/(q^5-1)/(q^3-1)/(q^2-1)/(q-1) The first, 41, coefficients are [1, 1, 2, 3, 6, 8, 13, 18, 27, 36, 51, 67, 92, 118, 156, 198, 256, 319, 404, 498, 620, 755, 926, 1116, 1353, 1615, 1935, 2291, 2720, 3194, 3759, 4384, 5120, 5932, 6879, 7923, 9131, 10458, 11981, 13654, 15561] For the, 2, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5, 6, 7, 8, 9}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}] the generating function for its P-partitions is (q^8+1)/(q^3-1)/(q^12-1)/(q^11-1)/(q^10-1)/(q^9-1)/(q^4-1)/(q^7-1)/(q^6-1)/(q^5 -1)/(q^2-1)/(q-1)^2/(q^3+q^2+q+1) The first, 41, coefficients are [1, 1, 2, 3, 6, 8, 13, 18, 28, 38, 55, 74, 105, 138, 188, 246, 328, 422, 552, 703, 905, 1140, 1446, 1804, 2263, 2796, 3467, 4251, 5222, 6351, 7735, 9341, 11288, 13542, 16246, 19370, 23090, 27369, 32425, 38232, 45044] For the, 3, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5, 6, 7, 8, 9}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}] the generating function for its P-partitions is (q^8-q^4+1)/(q^3-1)/(q^7-1)/(q^4-1)^2/(q^15-1)/(q^14-1)/(q^13-1)/(q^12-1)/(q^11 -1)/(q^10-1)/(q^9-1)/(q^6-1)/(q^5-1)/(q^2-1)/(q-1)^2/(q^3+q^2+q+1) The first, 41, coefficients are [1, 1, 2, 3, 6, 8, 13, 18, 28, 38, 55, 74, 106, 140, 192, 253, 341, 442, 584, 751, 978, 1245, 1598, 2017, 2563, 3207, 4030, 5008, 6239, 7694, 9507, 11649, 14288, 17399, 21194, 25660, 31068, 37408, 45026, 53944, 64587] For the, 4, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5, 6, 7, 8, 9}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}] the generating function for its P-partitions is (q^8-q^4+1)*(q^16+1)/(q^3-1)/(q^7-1)/(q^11-1)/(q^20-1)/(q^19-1)/(q^18-1)/(q^17-\ 1)/(q^4-1)^2/(q^15-1)/(q^14-1)/(q^13-1)/(q^10-1)/(q^9-1)/(q^12-1)/(q^6-1)/(q^5-\ 1)/(q^2-1)/(q-1)^2/(q^3+q^2+q+1) The first, 41, coefficients are [1, 1, 2, 3, 6, 8, 13, 18, 28, 38, 55, 74, 106, 140, 192, 253, 342, 444, 588, 758, 991, 1265, 1630, 2065, 2636, 3312, 4182, 5221, 6540, 8107, 10074, 12414, 15320, 18768, 23009, 28037, 34176, 41429, 50218, 60590, 73078] For the, 5, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5, 6, 7, 8, 9}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} , {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23}] the generating function for its P-partitions is (q^16+1)*(q^16-q^12+q^8-q^4+1)/(q^5-1)/(q^9-1)/(q^13-1)/(q^17-1)/(q^12-1)^2/(q^ 23-1)/(q^22-1)/(q^21-1)/(q^20-1)/(q^19-1)/(q^18-1)/(q^4-1)^3/(q^15-1)/(q^14-1)/ (q^11-1)/(q^10-1)/(q^7-1)/(q^6-1)/(q^3-1)/(q^2-1)/(q-1) The first, 41, coefficients are [1, 1, 2, 3, 6, 8, 13, 18, 28, 38, 55, 74, 106, 140, 192, 253, 342, 444, 588, 758, 992, 1267, 1634, 2072, 2649, 3332, 4214, 5269, 6613, 8212, 10226, 12627, 15621, 19181, 23576, 28802, 35209, 42800, 52037, 62975, 76201] The whole thing took, 18.890, seconds of CPU time The Two-rowed, atomic object with, 4, cells in each row and the second row is jutting by, 3, cells compared to the one above, which in our notation is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}] The generating functions for up to, 6, rows i.e. up to the, 5, -th power of it are as follows This is the story of the P-partitions for the n-th power of the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}] glued by the first and last, 4, vetrices for n from 1 to, 5 as well as the first , 41, terms of its counting series For the, 1, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}] the generating function for its P-partitions is 1/(q-1)/(q^2-1)/(q^3-1)/(q^4-1)/(q^5-1)/(q^6-1)/(q^7-1)/(q^8-1) The first, 41, coefficients are [1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 40, 52, 70, 89, 116, 146, 186, 230, 288, 352, 434, 525, 638, 764, 919, 1090, 1297, 1527, 1801, 2104, 2462, 2857, 3319, 3828, 4417, 5066, 5812, 6630, 7564, 8588, 9749] For the, 2, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5, 6, 7, 8}, {1, 2, 3, 4, 5, 6, 7, 8, 9}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}] the generating function for its P-partitions is 1/(q-1)/(q^2-1)/(q^3-1)/(q^4-1)/(q^5-1)/(q^6-1)/(q^7-1)/(q^8-1)/(q^9-1)/(q^10-1 )/(q^11-1)/(q^12-1) The first, 41, coefficients are [1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 100, 133, 172, 224, 285, 366, 460, 582, 725, 905, 1116, 1380, 1686, 2063, 2503, 3036, 3655, 4401, 5262, 6290, 7476, 8877, 10489, 12384, 14552, 17084, 19978, 23334] For the, 3, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5, 6, 7, 8}, {1, 2, 3, 4, 5, 6, 7, 8, 9}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}] the generating function for its P-partitions is 1/(q-1)/(q^2-1)/(q^3-1)/(q^4-1)/(q^5-1)/(q^6-1)/(q^7-1)/(q^8-1)/(q^9-1)/(q^10-1 )/(q^11-1)/(q^12-1)/(q^13-1)/(q^14-1)/(q^15-1)/(q^16-1) The first, 41, coefficients are [1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 296, 383, 486, 620, 780, 983, 1225, 1530, 1891, 2339, 2871, 3523, 4293, 5231, 6334, 7665, 9228, 11098, 13287, 15892, 18928, 22518, 26694, 31603] For the, 4, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5, 6, 7, 8}, {1, 2, 3, 4, 5, 6, 7, 8, 9}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}] the generating function for its P-partitions is 1/(q-1)/(q^2-1)/(q^3-1)/(q^4-1)/(q^5-1)/(q^6-1)/(q^7-1)/(q^8-1)/(q^9-1)/(q^10-1 )/(q^11-1)/(q^12-1)/(q^13-1)/(q^14-1)/(q^15-1)/(q^16-1)/(q^17-1)/(q^18-1)/(q^19 -1)/(q^20-1) The first, 41, coefficients are [1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 791, 1000, 1251, 1568, 1946, 2417, 2980, 3673, 4498, 5507, 6703, 8154, 9871, 11937, 14375, 17293, 20722, 24803, 29588, 35251] For the, 5, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5, 6, 7, 8}, {1, 2, 3, 4, 5, 6, 7, 8, 9}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} , {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23}] the generating function for its P-partitions is 1/(q-1)/(q^2-1)/(q^3-1)/(q^4-1)/(q^5-1)/(q^6-1)/(q^7-1)/(q^8-1)/(q^9-1)/(q^10-1 )/(q^11-1)/(q^12-1)/(q^13-1)/(q^14-1)/(q^15-1)/(q^16-1)/(q^17-1)/(q^18-1)/(q^19 -1)/(q^20-1)/(q^21-1)/(q^22-1)/(q^23-1)/(q^24-1) The first, 41, coefficients are [1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1957, 2434, 3006, 3711, 4553, 5585, 6812, 8304, 10076, 12213, 14744, 17782, 21365, 25642, 30677, 36654] The whole thing took, 12.328, seconds of CPU time The Two-rowed, atomic object with, 5, cells in each row and the second row is jutting by, -4, cells compared to the one above, which in our notation is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {}, {6}, {6, 7}, {6, 7, 8}, {1, 6, 7, 8, 9}] The generating functions for up to, 6, rows i.e. up to the, 5, -th power of it are as follows This is the story of the P-partitions for the n-th power of the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {}, {6}, {6, 7}, {6, 7, 8}, {1, 6, 7, 8, 9}] glued by the first and last, 5, vetrices for n from 1 to, 5 as well as the first , 41, terms of its counting series For the, 1, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {}, {6}, {6, 7}, {6, 7, 8}, {1, 6, 7, 8, 9}] the generating function for its P-partitions is (1+q+14*q^8+11*q^7+q^24+18*q^15+3*q^22+7*q^20+9*q^19+20*q^13+14*q^17+16*q^16+18 *q^10+9*q^6+19*q^11+2*q^2+3*q^3+5*q^4+16*q^9+6*q^5+20*q^12+11*q^18+2*q^23+5*q^ 21+q^25+19*q^14)/(q-1)/(q^2-1)/(q^3-1)/(q^4-1)/(q^5-1)/(q^6-1)/(q^7-1)/(q^8-1)/ (q^9-1)/(q^10-1) The first, 41, coefficients are [1, 2, 5, 10, 20, 35, 62, 102, 166, 259, 398, 593, 873, 1256, 1785, 2494, 3447, 4697, 6340, 8458, 11188, 14654, 19048, 24550, 31428, 39936, 50436, 63285, 78968, 97968, 120931, 148503, 181527, 220860, 267590, 322832, 387983, 464477, 554082, 658630, 780342] For the, 2, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {}, {6}, {6, 7}, {6, 7, 8}, {1, 6, 7, 8, 9}, {}, {11}, {11, 12}, {11, 12, 13}, {6, 11, 12, 13, 14}] the generating function for its P-partitions is -(1+2*q+8752*q^53+30931*q^38+27832*q^33+4939*q^56+2445*q^59+17594*q^48+21448*q^ 46+25029*q^44+15669*q^49+29146*q^41+23072*q^30+30578*q^36+29938*q^35+30909*q^37 +155*q^8+97*q^7+11787*q^24+1812*q^15+8572*q^22+30036*q^40+29018*q^34+19534*q^47 +23294*q^45+13793*q^50+5918*q^20+4813*q^19+26608*q^43+10311*q^52+1002*q^13+3049 *q^17+2373*q^16+353*q^10+60*q^6+30634*q^39+6060*q^55+3139*q^58+11999*q^51+7331* q^54+1871*q^60+510*q^11+5*q^2+10*q^3+20*q^4+236*q^9+34*q^5+725*q^12+3863*q^18+ 10111*q^23+7167*q^21+3969*q^57+17352*q^27+24819*q^31+26424*q^32+27994*q^42+ 13565*q^25+21212*q^29+1362*q^14+15435*q^26+19296*q^28+367*q^65+1408*q^61+101*q^ 68+751*q^63+161*q^67+530*q^64+1038*q^62+246*q^66+5*q^73+62*q^69+36*q^70+2*q^74+ 10*q^72+20*q^71+q^75)/(q^15-1)/(q^14-1)/(q^13-1)/(q^12-1)/(q^8-1)/(q^11-1)/(q^ 10-1)/(q^9-1)/(q^7-1)/(q^6-1)/(q-1)^3/(q^5-1)/(q^3-1)/(q+1)^2/(q^2+1) The first, 41, coefficients are [1, 3, 9, 22, 51, 106, 214, 407, 751, 1334, 2310, 3889, 6412, 10341, 16383, 25498, 39084, 59021, 87966, 129445, 188309, 270930, 385876, 544275, 760825, 1054399, 1449531, 1977415, 2678031, 3601722, 4812241, 6389131, 8432031, 11064227, 14438631, 18742961, 24208003, 31114978, 39806840, 50698648, 64292809] For the, 3, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {}, {6}, {6, 7}, {6, 7, 8}, {1, 6, 7, 8, 9}, {}, {11}, {11, 12}, {11, 12, 13}, {6, 11, 12, 13, 14}, {}, {16}, {16, 17}, {16, 17, 18}, {11, 16, 17, 18, 19}] the generating function for its P-partitions is (1+3*q+105968467*q^53+13383950*q^38+4971193*q^33+51792*q^133+139308389*q^56+ 175542235*q^59+60875546*q^48+47039696*q^46+35568980*q^44+35120*q^134+68707046*q ^49+536326*q^126+152106*q^130+212109*q^129+75225*q^132+107690*q^131+22414760*q^ 41+398060*q^127+2055335*q^121+715139*q^125+2515260*q^30+9192677*q^36+7542214*q^ 35+11128655*q^37+728*q^8+398*q^7+512025*q^24+22217*q^15+278454*q^22+18991717*q^ 40+4146144*q^118+6145222*q^34+3307613*q^119+53654924*q^47+41017166*q^45+ 77147460*q^50+48183631*q^104+144671*q^20+54909491*q^103+102339*q^19+30671426*q^ 43+97607999*q^98+141448884*q^94+62242451*q^102+70186842*q^101+87886591*q^99+ 78738969*q^100+6214*q^138+177794961*q^91+153362419*q^93+165507989*q^92+ 118636238*q^96+95802909*q^52+31505598*q^107+9363*q^13+49156*q^17+33324*q^16+ 2189*q^10+211*q^6+15993154*q^39+127776452*q^55+163275937*q^58+86185987*q^51+ 116643050*q^54+187869996*q^60+3635*q^11+1600041*q^122+944161*q^124+9*q^2+22*q^3 +51*q^4+16492059*q^111+7809079*q^115+9508120*q^114+1279*q^9+5157831*q^117+ 6369235*q^116+237693096*q^86+276958082*q^82+284649088*q^81+291298634*q^80+ 42053575*q^105+23068649*q^109+2617992*q^120+258818196*q^84+304278706*q^77+ 301172588*q^78+296828495*q^79+13815198*q^112+11498853*q^113+36501790*q^106+105* q^5+190124593*q^90+202391669*q^89+226291417*q^87+268314995*q^83+248573772*q^85+ 214485386*q^88+1339*q^141+27037985*q^108+19564799*q^110+51*q^146+5904*q^12+ 71443*q^18+379631*q^23+201905*q^21+129849406*q^95+107871796*q^97+151168239*q^57 +1182457*q^27+3181185*q^31+3991926*q^32+26297012*q^42+683425*q^25+1972554*q^29+ 14564*q^14+903253*q^26+1533973*q^28+217*q^144+758*q^142+22*q^147+23423*q^135+ 413*q^143+2296*q^140+246620960*q^65+200154978*q^61+275438403*q^68+224153553*q^ 63+266631374*q^67+235637014*q^64+212287468*q^62+256989852*q^66+306109061*q^76+ 303802163*q^73+283310941*q^69+290157867*q^70+305869315*q^74+300465124*q^72+ 295898733*q^71+306641459*q^75+1234719*q^123+292245*q^128+9*q^148+3*q^149+108*q^ 145+q^150+3823*q^139+15351*q^136+9867*q^137)/(q^20-1)/(q^19-1)/(q^18-1)/(q^15-1 )/(q^17-1)/(q^16-1)/(q^14-1)/(q^13-1)/(q^4-1)/(q^8-1)/(q-1)^2/(q^7-1)/(q^12-1)/ (q^11-1)/(q^10-1)/(q^3-1)/(q^9-1)/(q^2-1)/(q^6-1)/(q^4+q^3+q+q^2+1) The first, 41, coefficients are [1, 4, 14, 40, 105, 249, 561, 1193, 2438, 4791, 9129, 16889, 30481, 53738, 92813, 157242, 261808, 428859, 692084, 1101286, 1729748, 2683669, 4116087, 6244805, 9377995, 13947134, 20552832, 30023656, 43496158, 62517197, 89180652, 126301265, 177643339, 248209943, 344618207, 475571117, 652462586, 890133468, 1207831025, 1630400658, 2189784044] For the, 4, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {}, {6}, {6, 7}, {6, 7, 8}, {1, 6, 7, 8, 9}, {}, {11}, {11, 12}, {11, 12, 13}, {6, 11, 12, 13, 14}, {}, {16}, {16, 17}, {16, 17, 18}, {11, 16, 17, 18, 19}, {}, {21}, {21, 22}, {21, 22, 23}, {16, 21, 22, 23, 24}] the generating function for its P-partitions is -(126256796*q^218+92239273*q^219+66861525*q^220+48072458*q^221+171522877*q^217+ 34271000*q^222+24216143*q^223+16953403*q^224+2439658*q^229+658515*q^232+1033225 *q^231+5477272*q^227+11754148*q^225+154974*q^235+255327*q^234+1598318*q^230+ 231333602*q^216+309825559*q^215+715395866*q^212+53875*q^237+30753*q^238+17128*q ^239+9288*q^240+92315*q^236+1+4*q+21429165297*q^53+675759533*q^38+161342999*q^ 33+10811220790752*q^133+37816645030*q^56+64437249282*q^59+7644731393*q^48+ 4903782770*q^46+3084878322*q^44+10614955850585*q^134+9478370460*q^49+ 11559546402741*q^126+11272727993908*q^130+11380977890893*q^129+10986965896033*q ^132+11141118238669*q^131+1480562803*q^41+11524839584780*q^127+11356851596558*q ^121+11569092586706*q^125+62748968*q^30+388658992*q^36+291915867*q^35+514102165 *q^37+2397*q^8+1179*q^7+7545323*q^24+145557*q^15+3438113*q^22+1146573757*q^40+ 10946201942547*q^118+217780931*q^34+11105690770325*q^119+6137308492*q^47+ 3899146580*q^45+11698757374*q^50+7085183099282*q^104+1494569*q^20+6753522377417 *q^103+966483*q^19+2428166142*q^43+5132930161230*q^98+3950265322013*q^94+ 6422620031619*q^102+6093851917631*q^101+5447834972258*q^99+5768517996505*q^100+ 9649845873519*q^138+3164557999148*q^91+3677751012309*q^93+3415705950019*q^92+ 4524488772790*q^96+17588693630*q^52+8070003351052*q^107+50821*q^13+387226*q^17+ 239461*q^16+8859*q^10+557*q^6+882840960*q^39+31419083892*q^55+54153045471*q^58+ 14375579644*q^51+26000367700*q^54+76393091762*q^60+16262*q^11+11446961477427*q^ 122+11553417852926*q^124+14*q^2+40*q^3+105*q^4+9300905872038*q^111+ 10344203181015*q^115+10106385000814*q^114+4681*q^9+10765371823559*q^117+ 10564301842317*q^116+2080658950779*q^86+1420437078164*q^82+1282627551929*q^81+ 1155062166763*q^80+7416158879589*q^105+8702546781933*q^109+11242857484526*q^120 +1728186522207*q^84+829757555022*q^77+929060010759*q^78+1037346953344*q^79+ 9583240400352*q^112+9852241837420*q^113+7744948501615*q^106+248*q^5+ 2924637784827*q^90+2696182166569*q^89+2274169586212*q^87+1568850428046*q^83+ 1898716869821*q^85+2479338518651*q^88+8777229682849*q^141+8389738875275*q^108+ 9006807958405*q^110+7165509331580*q^146+29100*q^12+616348*q^18+5121731*q^23+ 2280913*q^21+4232721068646*q^95+4824836492922*q^97+45340042667*q^57+22682019*q^ 27+86629568*q^31+118669466*q^32+1901208014*q^42+10998390*q^25+45083893*q^29+ 86879*q^14+15870821*q^26+32119379*q^28+7824171458542*q^144+8466328263609*q^142+ 6833853167066*q^147+10399356525383*q^135+8148104747907*q^143+9079189093388*q^ 140+169684792364*q^65+90240750097*q^61+263080031164*q^68+124595628776*q^63+ 228040635648*q^67+145648386003*q^64+106220954869*q^62+197032426600*q^66+ 738979727601*q^76+513065361876*q^73+302541615319*q^69+346835592194*q^70+ 581110245819*q^74+451647488784*q^72+396389750889*q^71+656255998823*q^75+ 11512621515173*q^123+11465191406202*q^128+47248406556*q^193+110276259476*q^188+ 22401367781*q^197+18401439598*q^198+15052049890*q^199+12259283332*q^200+ 27158323629*q^196+9940702446*q^201+8024286955*q^202+6447416591*q^203+4103096039 *q^205+2005798957*q^208+933820656*q^211+1211737553*q^210+2559545624*q^207+ 3249005442*q^206+6502610172690*q^148+5155906072*q^204+412157530*q^214+544722557 *q^213+1563359655*q^209+6173173703464*q^149+7496120096916*q^145+4898620022999*q ^153+5846863157432*q^150+5524916369376*q^151+5208483148126*q^152+4596286170202* q^154+3742539656576*q^157+2981362475257*q^160+3477891471384*q^158+3224047723802 *q^159+4017540198909*q^156+4302340551012*q^155+2322402956594*q^163+ 2750095089487*q^161+2126063337584*q^164+1941322923944*q^165+1768040537102*q^166 +2530414298940*q^162+1606012750994*q^167+1454980457116*q^168+1314635519574*q^ 169+1064569726408*q^171+759829944553*q^174+528580178663*q^177+598287001005*q^ 176+852618918252*q^173+954046626475*q^172+1184627403793*q^170+675211105389*q^ 175+465615442095*q^178+312533833156*q^181+408923726951*q^179+271956738824*q^182 +235899469441*q^183+203966109697*q^184+358045519363*q^180+175780900219*q^185+ 150989173090*q^186+129257954838*q^187+93755109113*q^189+56389093669*q^192+ 32792489099*q^195+39438867680*q^194+67047009333*q^191+79427308401*q^190+ 9370592535520*q^139+10165709214384*q^136+9915388861883*q^137+8066831*q^226+ 413378*q^233+3677321*q^228+4885*q^241+2485*q^242+1215*q^243+569*q^244+252*q^245 +105*q^246+4*q^249+14*q^248+q^250+40*q^247)/(q^24-1)/(q^25-1)/(q^19-1)/(q^20-1) /(q^21-1)/(q^22-1)/(q^23-1)/(q^14-1)/(q^15-1)/(q^16-1)/(q^18-1)/(q^17-1)/(q^3-1 )/(q^9-1)/(q^4-1)/(q^8-1)/(q-1)^3/(q^7-1)/(q^13-1)/(q^6-1)/(q^12-1)/(q^11-1)/(q ^5+2*q^4+2*q^3+2*q^2+2*q+1)/(q^10-1) The first, 41, coefficients are [1, 5, 20, 65, 190, 502, 1244, 2904, 6475, 13850, 28617, 57295, 111607, 212009, 393761, 716301, 1278573, 2242398, 3869324, 6575931, 11018491, 18218095, 29747520, 48003214, 76603165, 120957515, 189088339, 292789258, 449264966, 683420563, 1031052564, 1543250931, 2292456247, 3380721944, 4950950125, 7202057027, 10409375452, 14951922945, 21348723046, 30306904517, 42785181101 ] For the, 5, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {}, {6}, {6, 7}, {6, 7, 8}, {1, 6, 7, 8, 9}, {}, {11}, {11, 12}, {11, 12, 13}, {6, 11, 12, 13, 14}, {}, {16}, {16, 17}, {16, 17, 18}, {11, 16, 17, 18, 19}, {}, {21}, {21, 22}, {21, 22, 23}, {16, 21, 22, 23, 24}, {}, {26}, {26, 27}, {26, 27, 28}, {21, 26, 27, 28, 29}] the generating function for its P-partitions is (693631638511931213*q^218+666647941028957631*q^219+639862724935444935*q^220+ 613335139040704040*q^221+720751667129292299*q^217+587121152046679662*q^222+ 561273399309958928*q^223+535841055216223318*q^224+416378555799319801*q^229+ 352038001343904429*q^232+372814332824699602*q^231+462474338359002107*q^227+ 510869731143488020*q^225+293918243754620570*q^235+312578701511744347*q^234+ 394267806335663586*q^230+747943078851458113*q^216+775138355920070116*q^215+ 856039093988835512*q^212+258767895543774562*q^237+242281825544139576*q^238+ 226519701169224095*q^239+211476674465745982*q^240+275980188603934913*q^236+1+5* q+1189381612111*q^53+14812593645*q^38+2575520230*q^33+179564159176705074*q^133+ 2524448504902*q^56+5175255749937*q^59+311912077450*q^48+176852600365*q^46+ 98307794331*q^44+192982940573542549*q^134+411300992910*q^49+103987894943692316* q^126+143372491947820185*q^130+132612054991850665*q^129+166832341174291719*q^ 132+154773614260279943*q^131+39157548681*q^41+112939464165503277*q^127+ 67227200074902598*q^121+95598514033095989*q^125+825395979*q^30+7509750103*q^36+ 5293921058*q^35+10581413308*q^37+6409*q^8+2884*q^7+66714814*q^24+681944*q^15+ 26527684*q^22+28489272770*q^40+50770280796614175*q^118+3705908024*q^34+ 55841364623641960*q^119+235433266768*q^47+132191976666*q^45+539887455299*q^50+ 11210030618681611*q^104+10038785*q^20+9930526953503322*q^103+6050779*q^19+ 72727064645*q^43+5265987737427406*q^98+3060291950212623*q^94+8780672744173099*q ^102+7749344473426403*q^101+6001363120251890*q^99+6826141204709448*q^100+ 253738404156332494*q^138+1993533249263214*q^91+2658426192168972*q^93+ 2304527076785286*q^92+4030586148524421*q^96+917987859860*q^52+15948881942978257 *q^107+204641*q^13+2101071*q^17+1207912*q^16+28087*q^10+1239*q^6+20605395690*q^ 39+1972171857715*q^55+4089061578364*q^58+705525854908*q^51+1534651555478*q^54+ 6526382292221*q^60+55951*q^11+73586481382897402*q^122+87749824583791096*q^124+ 20*q^2+65*q^3+190*q^4+24889853636481657*q^111+37783013429140475*q^115+ 34125729343722219*q^114+13654*q^9+46084469395503281*q^117+41762406162169364*q^ 116+934739386954001*q^86+489570511575575*q^82+414018303640363*q^81+ 349271547123818*q^80+12631048537484976*q^105+19994263386780007*q^109+ 61319613362697770*q^120+679650200116281*q^84+206571427194027*q^77+ 246720421117138*q^78+293920910233010*q^79+27698171900581424*q^112+ 30770568163489206*q^113+14206202272420553*q^106+501*q^5+1720831273647873*q^90+ 1482226522458437*q^89+1092454230117595*q^87+577518121071398*q^83+ 797974685488425*q^85+1273914711875890*q^88+306891464766421823*q^141+ 17873253843287700*q^108+22327632881930665*q^110+409652610000662834*q^146+108387 *q^12+3593732*q^18+42315818*q^23+16425808*q^21+3515663203143680*q^95+ 4611644215700094*q^97+3218946561673*q^57+245083840*q^27+1215851494*q^31+ 1776517408*q^32+53513356493*q^42+104018476*q^25+555607892*q^29+377570*q^14+ 160473633*q^26+370716958*q^28+366489211776765593*q^144+326051790480321099*q^142 +432207311287332407*q^147+207100760473864647*q^135+345921671440077184*q^143+ 288449312090351438*q^140+19768452627950*q^65+8201223298911*q^61+36971536454846* q^68+12817772683563*q^63+30100980676242*q^67+15944103539674*q^64+10270269553915 *q^62+24431912106850*q^66+172507743021364*q^76+98873646250898*q^73+ 45273324421796*q^69+55275008660936*q^70+119352789118099*q^74+81681997468622*q^ 72+67289840564833*q^71+143681947665237*q^75+80419891098067855*q^123+ 122474130007905767*q^128+1225798519652149819*q^193+1248110000554741557*q^188+ 1180974975194799572*q^197+1166317103162820435*q^198+1150376030202801754*q^199+ 1133208029914949170*q^200+1194297468832665590*q^196+1114873149133837649*q^201+ 1095434866517194194*q^202+1074959735277038966*q^203+1031178279942240708*q^205+ 959528500354855253*q^208+882532526187118441*q^211+908663249193594617*q^210+ 984108390704846792*q^207+1008017052947864554*q^206+455377208000714977*q^148+ 1053517012795425481*q^204+802267662402840014*q^214+829259118171236763*q^213+ 934354343985611829*q^209+479132589998960109*q^149+387739120516519231*q^145+ 579284702683773026*q^153+503440028129163816*q^150+528262371209043370*q^151+ 553558766238434300*q^152+605392081511719860*q^154+685470213006959587*q^157+ 766930067603962488*q^160+712554437201202309*q^158+739729974731934309*q^159+ 658541418850183198*q^156+631829309461147098*q^155+847975770968321671*q^163+ 794085560998570261*q^161+874562705198510910*q^164+900810111362354561*q^165+ 926641266696090753*q^166+821125170519046486*q^162+951978945776628024*q^167+ 976745791021832811*q^168+1000864693048259817*q^169+1046853801740883650*q^171+ 1109107744579486899*q^174+1161627925400657986*q^177+1145309899326125247*q^176+ 1089349188387247310*q^173+1068574540580529871*q^172+1024259178361609585*q^170+ 1127782797721407219*q^175+1176679422961152755*q^178+1213722040378998858*q^181+ 1190410938838529289*q^179+1223217379679586657*q^182+1231224739178629911*q^183+ 1237714805540349430*q^184+1202773328202982478*q^180+1242663733255207431*q^185+ 1246053286866933678*q^186+1247870950037089634*q^187+1246769550681677459*q^189+ 1233349705694247395*q^192+1206236804606599678*q^195+1216749879480276346*q^194+ 1239375768220522593*q^191+1243854552499858774*q^190+270730803049316783*q^139+ 221927713488408521*q^136+237471731526531384*q^137+486401398779003539*q^226+ 331955153197533889*q^233+439123111210131222*q^228+75576189418851957*q^253+ 63035840097775249*q^255+3187689733287884*q^281+835533288758395*q^290+ 308896024259641*q^296+434566547445699*q^294+1142511105024319*q^288+ 978149745579313*q^289+181646117852128*q^299+125840221902621*q^301+ 104317182866962*q^302+217373590300570*q^298+86236778744910*q^303+71090084602153 *q^304+58436497858513*q^305+47895646483052*q^306+16927050693602*q^311+ 20972184621143*q^310+13617775887752*q^312+5514368657551*q^316+8725764842040*q^ 314+1273026792914*q^322+756285232702*q^324+579166865268*q^325+1641338920342*q^ 321+190287146775*q^329+983289594380*q^323+105929756015*q^331+441557416058*q^326 +335108340127*q^327+78421511361*q^332+57743696488*q^333+16025202495*q^337+ 1927515686*q^343+896009198*q^345+603221211*q^346+2793430084*q^342+112870396*q^ 350+1319581205*q^344+45864857*q^352+142338083651*q^330+253130579335*q^328+ 42281867810*q^334+22278508654*q^336+5736755415*q^340+11454351181*q^338+ 2107686689016*q^320+8133768555*q^339+151392091237168*q^300+25901240043475*q^309 +31888801426756*q^308+39140096397841*q^307+10919180572838*q^313+6948899146052*q ^315+2695886113819*q^319+4360234257852*q^317+402501036*q^347+72357337*q^351+ 28728190*q^353+266078154*q^348+174186728*q^349+17769504*q^354+10845802*q^355+ 1294303*q^359+29341*q^365+6609*q^367+2954*q^368+58783*q^364+65*q^372+14179*q^ 366+5*q^374+6526946*q^356+2257184*q^358+217029*q^362+728553*q^360+401984*q^361+ 30782796202*q^335+4017783534*q^341+1259*q^369+114433*q^363+190*q^371+3869255*q^ 357+506*q^370+20*q^373+q^375+197145501004906419*q^241+183516718486574817*q^242+ 170578828363314539*q^243+158318479001936496*q^244+146720648971458050*q^245+ 135768829126491007*q^246+106605769676546251*q^249+115730819060479129*q^248+ 98049349359813022*q^250+125445202243280942*q^247+69076765417803500*q^254+ 90040217933324885*q^251+3434975806286*q^318+47440952648711134*q^258+ 52240051599752007*q^257+38932182609372697*q^260+31737028446035222*q^262+ 28582037076463623*q^263+43012071217128079*q^259+57431016853646797*q^256+ 25696686950833858*q^264+23062777763704314*q^265+20662836101920757*q^266+ 18480138249339127*q^267+10293960223032181*q^272+11614274031318870*q^271+ 9106774168812070*q^273+5473186245237588*q^277+7087128855406871*q^275+ 82556552004733156*q^252+35180558205166029*q^261+14703417370686046*q^269+ 16498726202027813*q^268+13079808461242767*q^270+8041369559986697*q^274+ 4795695048289678*q^278+3659985755865141*q^280+2770644463389955*q^282+ 2403161765663301*q^283+4193737191900191*q^279+6234142592666065*q^276+ 2080040040061366*q^284+1796534569716596*q^285+1548328235868694*q^286+ 1331502827750695*q^287+605433859625721*q^292+712069162073657*q^291+ 513550321590725*q^293+259455611336180*q^297+366835460054597*q^295)/(q^30-1)/(q^ 29-1)/(q^28-1)/(q^27-1)/(q^26-1)/(q^25-1)/(q^24-1)/(q^23-1)/(q^22-1)/(q^19-1)/( q^20-1)/(q^21-1)/(q^18-1)/(q^17-1)/(q^8+q^7-q-1)/(q^14-1)/(q^15-1)/(q^16-1)/(q^ 3-1)^2/(q^9-1)/(q^4-1)^2/(q^8-1)/(q-1)^4/(q^13-1)/(q^10-q^8+q^6-q^4+q^2-1)/(q^3 +1)/(q^10+q^9+q^8+q^7+q^6+q^5+q^4+q^3+q^2+q+1)/(q^8+q^6+q^4+q^2+1)/(q^4+q^3+q+q ^2+1) The first, 41, coefficients are [1, 6, 27, 98, 315, 913, 2461, 6230, 14996, 34538, 76610, 164323, 342178, 693691, 1372698, 2656826, 5038896, 9378945, 17156118, 30877511, 54737806, 95666530, 164979088, 280945735, 472759870, 786600360, 1294826527, 2109791918, 3404453237, 5442859404, 8624960158, 13551965614, 21121088576, 32661946725, 50131791337, 76393200757, 115606893495, 173784215131, 259560176313, 385269375374, 568436177702] The whole thing took, 7144.043, seconds of CPU time The Two-rowed, atomic object with, 5, cells in each row and the second row is jutting by, -3, cells compared to the one above, which in our notation is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {}, {6}, {6, 7}, {1, 6, 7, 8}, {1, 2, 6, 7, 8, 9}] The generating functions for up to, 6, rows i.e. up to the, 5, -th power of it are as follows This is the story of the P-partitions for the n-th power of the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {}, {6}, {6, 7}, {1, 6, 7, 8}, {1, 2, 6, 7, 8, 9}] glued by the first and last, 5, vetrices for n from 1 to, 5 as well as the first , 41, terms of its counting series For the, 1, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {}, {6}, {6, 7}, {1, 6, 7, 8}, {1, 2, 6, 7, 8, 9}] the generating function for its P-partitions is (q^20+q^19+2*q^18+3*q^17+5*q^16+6*q^15+8*q^14+9*q^13+11*q^12+11*q^11+12*q^10+11 *q^9+10*q^8+8*q^7+7*q^6+5*q^5+4*q^4+3*q^3+2*q^2+q+1)/(q^6+q^5-q-1)/(q^9-1)/(q^8 -1)/(q^7-1)/(q^6-1)/(q^5-1)/(q^4-1)/(q^3-1)/(q-1)^2 The first, 41, coefficients are [1, 2, 5, 10, 19, 34, 59, 97, 157, 245, 375, 559, 821, 1181, 1677, 2344, 3237, 4413, 5955, 7947, 10512, 13774, 17905, 23087, 29560, 37577, 47468, 59584, 74368, 92297, 113962, 139995, 171176, 208339, 252490, 304718, 366315, 438676, 523448, 622405, 737618] For the, 2, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {}, {6}, {6, 7}, {1, 6, 7, 8}, {1, 2, 6, 7, 8, 9}, {}, {11}, {11, 12}, {6, 11, 12, 13}, {6, 7, 11, 12, 13, 14}] the generating function for its P-partitions is -(1+q+282*q^38+603*q^33+10*q^48+25*q^46+55*q^44+5*q^49+137*q^41+776*q^30+405*q^ 36+469*q^35+338*q^37+45*q^8+30*q^7+820*q^24+295*q^15+734*q^22+180*q^40+540*q^34 +15*q^47+36*q^45+3*q^50+618*q^20+549*q^19+75*q^43+q^52+192*q^13+417*q^17+356*q^ 16+87*q^10+21*q^6+225*q^39+q^51+115*q^11+3*q^2+5*q^3+8*q^4+63*q^9+13*q^5+153*q^ 12+486*q^18+779*q^23+676*q^21+853*q^27+722*q^31+670*q^32+105*q^42+841*q^25+811* q^29+243*q^14+858*q^26+842*q^28)/(q^13-1)/(q^8+q^7-q-1)/(q^7+q^6+q^5-q^2-q-1)/( q^6-1)/(q^4-1)^2/(q^12-1)/(q^11-1)/(q^10-1)/(q^9-1)/(q^5-q^4+q-1)/(q-1)^3/(q^5-\ 1) The first, 41, coefficients are [1, 3, 9, 22, 49, 102, 202, 381, 695, 1226, 2106, 3527, 5783, 9290, 14662, 22755, 34785, 52424, 77988, 114600, 166503, 239329, 340584, 480103, 670781, 929294, 1277207, 1742085, 2359123, 3172836, 4239427, 5629273, 7430345, 9751807, 12728840, 16527815, 21353012, 27453896, 35134420, 44763107, 56785752] For the, 3, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {}, {6}, {6, 7}, {1, 6, 7, 8}, {1, 2, 6, 7, 8, 9}, {}, {11}, {11, 12}, {6, 11, 12, 13}, {6, 7, 11, 12, 13, 14}, {}, {16}, {16, 17}, {11, 16, 17, 18}, {11, 12, 16, 17, 18, 19}] the generating function for its P-partitions is (1+2*q+2246794*q^53+888535*q^38+452664*q^33+2249884*q^56+2120837*q^59+1960355*q ^48+1770269*q^46+1555114*q^44+2041872*q^49+1214489*q^41+273639*q^30+694723*q^36 +607129*q^35+788672*q^37+273*q^8+160*q^7+77895*q^24+5464*q^15+47166*q^22+ 1102484*q^40+526407*q^34+1869224*q^47+1664936*q^45+2112548*q^50+29*q^104+27255* q^20+56*q^103+20324*q^19+1442292*q^43+898*q^98+4832*q^94+107*q^102+190*q^101+ 551*q^99+332*q^100+13844*q^91+6978*q^93+9920*q^92+2179*q^96+2215978*q^52+2*q^ 107+2572*q^13+10856*q^17+7766*q^16+719*q^10+92*q^6+993405*q^39+2263917*q^55+ 2177672*q^58+2170868*q^51+2262958*q^54+2051657*q^60+1122*q^11+6*q^2+13*q^3+26*q ^4+448*q^9+59115*q^86+151015*q^82+185845*q^81+226504*q^80+13*q^105+96611*q^84+ 387243*q^77+326858*q^78+273339*q^79+6*q^106+50*q^5+19046*q^90+25787*q^89+45406* q^87+121414*q^83+75993*q^85+34461*q^88+q^108+1719*q^12+14965*q^18+60945*q^23+ 36065*q^21+3273*q^95+1412*q^97+2220874*q^57+152554*q^27+326426*q^31+386064*q^32 +1328294*q^42+98458*q^25+227305*q^29+3785*q^14+123191*q^26+187117*q^28+1567529* q^65+1971144*q^61+1225431*q^68+1782452*q^63+1339883*q^67+1677483*q^64+1880968*q ^62+1454515*q^66+454824*q^76+700113*q^73+1112419*q^69+1002383*q^70+611426*q^74+ 795330*q^72+896294*q^71+529529*q^75)/(q^12+q^10-q^2-1)/(q-1)^4/(q^12+q^9-q^3-1) /(q^17-1)/(q^16-1)/(q^15-1)/(q^14-1)/(q^13-1)/(q^6+q^5-q-1)/(q^4-1)/(q^8-1)/(q^ 10-q^8+q^6-q^4+q^2-1)/(q^11-1)/(q^3-1)^2/(q^9-1)/(q^7-1)/(q^4+q^3+q+q^2+1) The first, 41, coefficients are [1, 4, 14, 40, 102, 240, 531, 1115, 2247, 4366, 8225, 15072, 26955, 47149, 80843, 136095, 225288, 367166, 589810, 934753, 1462866, 2262368, 3460038, 5236328, 7846072, 11646112, 17132549, 24989876, 36156282, 51908943, 73975901, 104680777, 147130482, 205455756, 285119948, 393311412, 539442366, 735777818, 998228259, 1347341434, 1809542429] For the, 4, power which is the poset [{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {}, {6}, {6, 7}, {1, 6, 7, 8}, {1, 2, 6, 7, 8, 9}, {}, {11}, {11, 12}, {6, 11, 12, 13}, {6, 7, 11, 12, 13, 14}, {}, {16}, {16, 17}, {11, 16, 17, 18}, {11, 12, 16, 17, 18, 19}, {}, {21}, {21, 22}, {16, 21, 22, 23}, {16, 17, 21, 22, 23, 24}] the generating function for its P-partitions is -(1+q+39700482*q^53+3954258*q^38+1385912*q^33+1148014*q^133+55306188*q^56+ 74107447*q^59+20866008*q^48+15601695*q^46+11433132*q^44+904096*q^134+23954279*q ^49+5034926*q^126+2252015*q^130+2780079*q^129+1447535*q^132+1811521*q^131+ 6894274*q^41+4156250*q^127+12030028*q^121+6061758*q^125+681611*q^30+2649875*q^ 36+2148880*q^35+3246747*q^37+225*q^8+126*q^7+133479*q^24+5875*q^15+72153*q^22+ 5761271*q^40+19007605*q^118+1731622*q^34+16403691*q^119+18087043*q^47+13389631* q^45+27369920*q^50+91308952*q^104+37436*q^20+98864208*q^103+26491*q^19+9710976* q^43+138505765*q^98+169027374*q^94+106612066*q^102+114500927*q^101+130504671*q^ 99+122485393*q^100+320843*q^138+188538055*q^91+175959564*q^93+182486048*q^92+ 154192040*q^96+35232849*q^52+70162868*q^107+2533*q^13+12816*q^17+8748*q^16+631* q^10+71*q^6+4786697*q^39+49737433*q^55+67505680*q^58+31124776*q^51+44535684*q^ 54+81018319*q^60+1021*q^11+10220943*q^122+7256356*q^124+5*q^2+9*q^3+19*q^4+ 46513363*q^111+28698459*q^115+32608117*q^114+378*q^9+21912616*q^117+25138775*q^ 116+209838606*q^86+213820375*q^82+212849353*q^81+211104137*q^80+83982228*q^105+ 57627931*q^109+14085132*q^120+213393000*q^84+201422189*q^77+205360271*q^78+ 208596748*q^79+41509986*q^112+36874959*q^113+76925019*q^106+38*q^5+194063218*q^ 90+199000048*q^89+206928452*q^87+214000081*q^83+211998736*q^85+203305354*q^88+ 134303*q^141+63725496*q^108+51888877*q^110+25468*q^146+1630*q^12+18559*q^18+ 98587*q^23+52222*q^21+161747496*q^95+146420897*q^97+61231804*q^57+313151*q^27+ 869675*q^31+1101870*q^32+8204935*q^42+178909*q^25+530063*q^29+3901*q^14+237792* q^26+409133*q^28+51321*q^144+98581*q^142+17548*q^147+706227*q^135+71457*q^143+ 181363*q^140+119020396*q^65+88206900*q^61+142944482*q^68+103287077*q^63+ 135002590*q^67+111096815*q^64+95644018*q^62+127009491*q^66+196829578*q^76+ 179595206*q^73+150767244*q^69+158411503*q^70+185861450*q^74+172890977*q^72+ 165806192*q^71+191623169*q^75+8635621*q^123+3410393*q^128+11977*q^148+8000*q^ 149+36326*q^145+1346*q^153+5299*q^150+3413*q^151+2185*q^152+830*q^154+154*q^157 +23*q^160+87*q^158+42*q^159+286*q^156+483*q^155+q^163+9*q^161+q^164+5*q^162+ 242235*q^139+547578*q^136+420805*q^137)/(q^5-1)/(q^16+q^12-q^4-1)/(q^20-1)/(q^ 19-1)/(q^21-1)/(q-1)^5/(q^12+q^11-q-1)/(q^16-1)/(q^15-1)/(q^14-1)/(q^18-1)/(q^ 17-1)/(q^3-1)/(q^9-1)/(q^4-1)^2/(q^7-1)/(q^13-1)/(q^6-1)/(q^2-1)/(q^9-q^8+q^7-q ^6+q^5-q^4+q^3-q^2+q-1)/(q^4+q^3+q+q^2+1) The first, 41, coefficients are [1, 5, 20, 65, 186, 486, 1184, 2724, 5982, 12621, 25728, 50881, 97960, 184097, 338503, 610123, 1079761, 1878887, 3218629, 5433773, 9049116, 14878171, 241