Let , M(n), be the Motzkin numbers. The following are (rigorously proved!) explicit expressions in terms of the Motzkin numbers M(n), for the number of 3-rowed n-celled Young-tableaux whose [i,j]-entry has m in it, for m between 1 and , 15, and the cell [i,j], with i<=3 (of course) and i*j<=m The number of three-rowed n-celled Young tableaux Y such that Y[1, 1] = 1, equals M(n) The first, 30, non-zero terms are [1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113634, 310572, 853467, 2356779, 6536382, 18199284, 50852019, 142547559, 400763223, 1129760415, 3192727797, 9043402501, 25669818476, 73007772802, 208023278209, 593742784829, 1697385471211, 4859761676391] The number of three-rowed n-celled Young tableaux Y such that Y[1, 2] = 2, equals M(n) - M(n - 1) The first, 30, non-zero terms are [1, 2, 5, 12, 30, 76, 196, 512, 1353, 3610, 9713, 26324, 71799, 196938, 542895, 1503312, 4179603, 11662902, 32652735, 91695540, 258215664, 728997192, 2062967382, 5850674704, 16626415975, 47337954326, 135015505407, 385719506620, 1103642686382, 3162376205180, 9073807670316] The number of three-rowed n-celled Young tableaux Y such that Y[2, 1] = 2, equals M(n - 1) The first, 30, non-zero terms are [1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113634, 310572, 853467, 2356779, 6536382, 18199284, 50852019, 142547559, 400763223, 1129760415, 3192727797, 9043402501, 25669818476, 73007772802, 208023278209, 593742784829, 1697385471211, 4859761676391] The number of three-rowed n-celled Young tableaux Y such that Y[1, 2] = 3, equals M(n - 1) - M(n - 3) The first, 30, non-zero terms are [1, 3, 7, 17, 42, 106, 272, 708, 1865, 4963, 13323, 36037, 98123, 268737, 739833, 2046207, 5682915, 15842505, 44315637, 124348275, 349911204, 987212856, 2791964574, 7913642086, 22477090679, 63964370301, 182353459733, 520735012027, 1489362193002, 4266018891562, 12236183875496] The number of three-rowed n-celled Young tableaux Y such that Y[1, 3] = 3, equals M(n) - 2 M(n - 1) + M(n - 3) The first, 30, non-zero terms are [1, 2, 5, 13, 34, 90, 240, 645, 1745, 4750, 13001, 35762, 98815, 274158, 763479, 2133396, 5979987, 16810230, 47379903, 133867389, 379085988, 1075754526, 3058710130, 8712773889, 24860863647, 71051135106, 203366046887, 582907674355, 1673014012178, 4807788778754, 13832711553880] The number of three-rowed n-celled Young tableaux Y such that Y[2, 1] = 3, equals M(n - 1) - M(n - 3) The first, 30, non-zero terms are [1, 3, 7, 17, 42, 106, 272, 708, 1865, 4963, 13323, 36037, 98123, 268737, 739833, 2046207, 5682915, 15842505, 44315637, 124348275, 349911204, 987212856, 2791964574, 7913642086, 22477090679, 63964370301, 182353459733, 520735012027, 1489362193002, 4266018891562, 12236183875496] The number of three-rowed n-celled Young tableaux Y such that Y[3, 1] = 3, equals M(n - 3) The first, 30, non-zero terms are [1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113634, 310572, 853467, 2356779, 6536382, 18199284, 50852019, 142547559, 400763223, 1129760415, 3192727797, 9043402501, 25669818476, 73007772802, 208023278209, 593742784829, 1697385471211] The number of three-rowed n-celled Young tableaux Y such that Y[1, 2] = 4, equals M(n - 3) The first, 30, non-zero terms are [1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113634, 310572, 853467, 2356779, 6536382, 18199284, 50852019, 142547559, 400763223, 1129760415, 3192727797, 9043402501, 25669818476, 73007772802, 208023278209, 593742784829, 1697385471211, 4859761676391] The number of three-rowed n-celled Young tableaux Y such that Y[1, 3] = 4, equals 2 M(n - 1) - 2 M(n - 2) - 2 M(n - 3) The first, 30, non-zero terms are [2, 6, 16, 42, 110, 290, 770, 2060, 5550, 15050, 41052, 112576, 310206, 858522, 2385480, 6652272, 18612246, 52232706, 146992512, 414727290, 1172899266, 3324408318, 9441828578, 26867376356, 76589103650, 218691373862, 625423467636, 1791238816346, 5137266840702, 14752844398210, 42418267505970] The number of three-rowed n-celled Young tableaux Y such that Y[1, 4] = 4, equals M(n) - 3 M(n - 1) + M(n - 2) + 2 M(n - 3) The first, 30, non-zero terms are [1, 2, 5, 13, 35, 95, 260, 715, 1975, 5476, 15236, 42527, 119055, 334218, 940656, 2653851, 7504107, 21263550, 60371133, 171722343, 489304893, 1396505971, 3991859600, 11427175469, 32756583281, 94020359956, 270195940537, 777394604005, 2239155358403, 6456289354775, 18634404895510] The number of three-rowed n-celled Young tableaux Y such that Y[2, 1] = 4, equals M(n - 1) - M(n - 2) - M(n - 3) The first, 30, non-zero terms are [1, 3, 8, 21, 55, 145, 385, 1030, 2775, 7525, 20526, 56288, 155103, 429261, 1192740, 3326136, 9306123, 26116353, 73496256, 207363645, 586449633, 1662204159, 4720914289, 13433688178, 38294551825, 109345686931, 312711733818, 895619408173, 2568633420351, 7376422199105, 21209133752985] The number of three-rowed n-celled Young tableaux Y such that Y[2, 2] = 4, equals 2 M(n - 2) - 2 M(n - 3) The first, 30, non-zero terms are [2, 4, 10, 24, 60, 152, 392, 1024, 2706, 7220, 19426, 52648, 143598, 393876, 1085790, 3006624, 8359206, 23325804, 65305470, 183391080, 516431328, 1457994384, 4125934764, 11701349408, 33252831950, 94675908652, 270031010814, 771439013240, 2207285372764, 6324752410360, 18147615340632] The number of three-rowed n-celled Young tableaux Y such that Y[3, 1] = 4, equals 2 M(n - 3) The first, 30, non-zero terms are [2, 4, 8, 18, 42, 102, 254, 646, 1670, 4376, 11596, 31022, 83670, 227268, 621144, 1706934, 4713558, 13072764, 36398568, 101704038, 285095118, 801526446, 2259520830, 6385455594, 18086805002, 51339636952, 146015545604, 416046556418, 1187485569658, 3394770942422, 9719523352782] The number of three-rowed n-celled Young tableaux Y such that Y[1, 3] = 5, equals 2 M(n - 2) + M(n - 3) - 5 M(n - 4) The first, 30, non-zero terms are [5, 12, 31, 78, 200, 518, 1358, 3596, 9609, 25880, 70191, 191548, 525603, 1449336, 4014165, 11162208, 31151055, 87221412, 244950717, 689813910, 1947546258, 5511399894, 15630730724, 44419400468, 126468351575, 360705236840, 1030469983857, 2948397336206, 8448194899848, 24239973013750, 69639690516918] The number of three-rowed n-celled Young tableaux Y such that Y[1, 4] = 5, equals 3 M(n - 1) - 6 M(n - 2) - 3 M(n - 3) + 6 M(n - 4) The first, 30, non-zero terms are [3, 9, 24, 66, 180, 492, 1347, 3699, 10191, 28173, 78147, 217473, 607077, 1699623, 4771503, 13430025, 37891881, 107151003, 303643962, 862171344, 2452616586, 6989138814, 19949419521, 57030566829, 163274157393, 468084277683, 1343676506844, 3861883516674, 11112438277116, 32011006046100, 92309192934537] The number of three-rowed n-celled Young tableaux Y such that Y[1, 5] = 5, equals M(n) - 4 M(n - 1) + 3 M(n - 2) + 3 M(n - 3) - 2 M(n - 4) The first, 30, non-zero terms are [1, 2, 5, 13, 35, 96, 266, 742, 2079, 5845, 16478, 46564, 131859, 374115, 1063350, 3027432, 8632923, 24654132, 70507689, 201914445, 578967109, 1662146662, 4777368962, 13746394338, 39595640825, 114167847976, 329502435057, 951860852845, 2752143262403, 7964069546810, 23064822388336] The number of three-rowed n-celled Young tableaux Y such that Y[2, 1] = 5, equals M(n - 1) - 2 M(n - 2) - M(n - 3) + 2 M(n - 4) The first, 30, non-zero terms are [1, 3, 8, 22, 60, 164, 449, 1233, 3397, 9391, 26049, 72491, 202359, 566541, 1590501, 4476675, 12630627, 35717001, 101214654, 287390448, 817538862, 2329712938, 6649806507, 19010188943, 54424719131, 156028092561, 447892168948, 1287294505558, 3704146092372, 10670335348700, 30769730978179] The number of three-rowed n-celled Young tableaux Y such that Y[2, 2] = 5, equals 3 M(n - 2) - 3 M(n - 3) The first, 30, non-zero terms are [6, 15, 36, 90, 228, 588, 1536, 4059, 10830, 29139, 78972, 215397, 590814, 1628685, 4509936, 12538809, 34988706, 97958205, 275086620, 774646992, 2186991576, 6188902146, 17552024112, 49879247925, 142013862978, 405046516221, 1157158519860, 3310928059146, 9487128615540, 27221423010948, 78206686288128] The number of three-rowed n-celled Young tableaux Y such that Y[3, 1] = 5, equals 3 M(n - 3) - M(n - 4) The first, 30, non-zero terms are [5, 10, 23, 54, 132, 330, 842, 2182, 5729, 15206, 40735, 109994, 299067, 818082, 2249829, 6216870, 17252367, 48061470, 134356773, 376790658, 1059742110, 2988518022, 8448422976, 23937479706, 67966052927, 193353499930, 551062061825, 1573205076278, 4498413628804, 12881899557962, 36940946363730] The number of three-rowed n-celled Young tableaux Y such that Y[1, 3] = 6, equals 5 M(n - 4) - 5 M(n - 6) The first, 30, non-zero terms are [5, 15, 35, 85, 210, 530, 1360, 3540, 9325, 24815, 66615, 180185, 490615, 1343685, 3699165, 10231035, 28414575, 79212525, 221578185, 621741375, 1749556020, 4936064280, 13959822870, 39568210430, 112385453395, 319821851505, 911767298665, 2603675060135, 7446810965010, 21330094457810, 61180919377480] The number of three-rowed n-celled Young tableaux Y such that Y[1, 4] = 6, equals 5 M(n - 2) - 4 M(n - 3) - 12 M(n - 4) + 6 M(n - 6) The first, 30, non-zero terms are [11, 27, 75, 203, 549, 1485, 4030, 10980, 30045, 82565, 227826, 631092, 1754505, 4894173, 13694994, 38432976, 108147177, 305078985, 862619943, 2444374179, 6940550763, 19744261691, 56267328150, 160617322584, 459203844907, 1314787952691, 3769695789087, 10822399715975, 31108314042693, 89523363085869, 257915154183800] The number of three-rowed n-celled Young tableaux Y such that Y[1, 5] = 6, equals 4 M(n - 1) - 12 M(n - 2) + 16 M(n - 4) - 4 M(n - 6) The first, 30, non-zero terms are [4, 12, 32, 88, 244, 680, 1900, 5320, 14924, 41944, 118104, 333172, 941612, 2665992, 7561488, 21482844, 61134252, 174243612, 497369760, 1421750016, 4069679932, 11664420296, 33473861356, 96175237568, 276637985236, 796579221724, 2296114228472, 6624990736192, 19133101370388, 55306464467024, 160007367250540] The number of three-rowed n-celled Young tableaux Y such that Y[1, 6] = 6, equals M(n) - 5 M(n - 1) + 6 M(n - 2) + 3 M(n - 3) - 6 M(n - 4) + M(n - 6) The first, 30, non-zero terms are [1, 2, 5, 13, 35, 96, 267, 749, 2114, 5992, 17038, 48566, 138712, 396852, 1137060, 3262212, 9370569, 26946786, 77572005, 223529605, 644726679, 1861263888, 5377928999, 15551831433, 45008351667, 130357629626, 377832295727, 1095895578355, 3180794204213, 9238206271580, 26848209630061] The number of three-rowed n-celled Young tableaux Y such that Y[2, 1] = 6, equals M(n - 1) - 3 M(n - 2) + 4 M(n - 4) - M(n - 6) The first, 30, non-zero terms are [1, 3, 8, 22, 61, 170, 475, 1330, 3731, 10486, 29526, 83293, 235403, 666498, 1890372, 5370711, 15283563, 43560903, 124342440, 355437504, 1017419983, 2916105074, 8368465339, 24043809392, 69159496309, 199144805431, 574028557118, 1656247684048, 4783275342597, 13826616116756, 40001841812635] The number of three-rowed n-celled Young tableaux Y such that Y[2, 2] = 6, equals 4 M(n - 2) - 8 M(n - 3) + 6 M(n - 4) - 6 M(n - 6) The first, 30, non-zero terms are [10, 30, 78, 202, 528, 1392, 3704, 9936, 26850, 73030, 199794, 549438, 1518030, 4211802, 11730294, 32783322, 91911294, 258429114, 728566242, 2059025526, 5832295092, 16555064764, 47083841340, 134154395916, 382893474494, 1094571135450, 3133720364610, 8984373008038, 25792357133712, 74137417680048, 213352211141392] The number of three-rowed n-celled Young tableaux Y such that Y[2, 3] = 6, equals 5 M(n - 3) - 10 M(n - 4) + 5 M(n - 6) The first, 30, non-zero terms are [5, 10, 25, 65, 170, 450, 1200, 3225, 8725, 23750, 65005, 178810, 494075, 1370790, 3817395, 10666980, 29899935, 84051150, 236899515, 669336945, 1895429940, 5378772630, 15293550650, 43563869445, 124304318235, 355255675530, 1016830234435, 2914538371775, 8365070060890, 24038943893770, 69163557769400] The number of three-rowed n-celled Young tableaux Y such that Y[3, 1] = 6, equals 4 M(n - 3) - 3 M(n - 4) - M(n - 6) The first, 30, non-zero terms are [9, 23, 55, 137, 346, 890, 2320, 6120, 16305, 43815, 118619, 323233, 885875, 2440317, 6753081, 18764619, 52334523, 146453445, 411097797, 1157210931, 3265899972, 9239082384, 26194663390, 74419305986, 211828907983, 604026391929, 1725231486213, 4935305757555, 14138867013722, 40561249572826, 116511765593000] The number of three-rowed n-celled Young tableaux Y such that Y[3, 2] = 6, equals 5 M(n - 6) The first, 30, non-zero terms are [5, 5, 10, 20, 45, 105, 255, 635, 1615, 4175, 10940, 28990, 77555, 209175, 568170, 1552860, 4267335, 11783895, 32681910, 90996420, 254260095, 712737795, 2003816115, 5648802075, 15963638985, 45217012505, 128349092380, 365038864010, 1040116391045, 2968713924145, 8486927356055] The number of three-rowed n-celled Young tableaux Y such that Y[1, 3] = 7, equals 5 M(n - 6) The first, 30, non-zero terms are [5, 10, 20, 45, 105, 255, 635, 1615, 4175, 10940, 28990, 77555, 209175, 568170, 1552860, 4267335, 11783895, 32681910, 90996420, 254260095, 712737795, 2003816115, 5648802075, 15963638985, 45217012505, 128349092380, 365038864010, 1040116391045, 2968713924145, 8486927356055, 24298808381955] The number of three-rowed n-celled Young tableaux Y such that Y[1, 4] = 7, equals 5 M(n - 3) + 6 M(n - 4) - 21 M(n - 5) - 6 M(n - 6) The first, 30, non-zero terms are [21, 63, 168, 446, 1180, 3140, 8405, 22645, 61385, 167355, 458661, 1263063, 3493443, 9700881, 27036345, 75601551, 212051103, 596446653, 1682013366, 4754770560, 13470888438, 38243831234, 108783661159, 309989955563, 884836424855, 2529671453701, 7242848585828, 20766383058558, 59618865187476, 171374431929180, 493194794938655] The number of three-rowed n-celled Young tableaux Y such that Y[1, 5] = 7, equals 9 M(n - 2) - 17 M(n - 3) - 21 M(n - 4) + 28 M(n - 5) + 11 M(n - 6) The first, 30, non-zero terms are [19, 47, 131, 364, 1016, 2831, 7892, 22018, 61519, 172192, 482921, 1357166, 3821997, 10785378, 30496071, 86394270, 245202009, 697148259, 1985418861, 5663299296, 16178674638, 46284927103, 132595331486, 380346454046, 1092358449905, 3140938698109, 9041440886197, 26054134761796, 75154472713544, 216996068366441, 627117174166216] The number of three-rowed n-celled Young tableaux Y such that Y[1, 6] = 7, equals 5 M(n - 1) - 20 M(n - 2) + 10 M(n - 3) + 30 M(n - 4) - 15 M(n - 5) - 10 M(n - 6) The first, 30, non-zero terms are [5, 15, 40, 110, 305, 855, 2410, 6820, 19350, 55010, 156640, 446660, 1275300, 3645660, 10433880, 29895360, 85750125, 246220695, 707716560, 2036223830, 5864208845, 16904314955, 48772673690, 140842326780, 407056323475, 1177410375505, 3408324561540, 9873743291990, 28624570942035, 83042665167805, 241078232735930] The number of three-rowed n-celled Young tableaux Y such that Y[1, 7] = 7, equals M(n) - 6 M(n - 1) + 10 M(n - 2) + M(n - 3) - 12 M(n - 4) + 3 M(n - 5) + 3 M(n - 6) The first, 30, non-zero terms are [1, 2, 5, 13, 35, 96, 267, 750, 2122, 6036, 17238, 49380, 141792, 407928, 1175436, 3391497, 9796761, 28327866, 81986293, 237481913, 688422119, 1997066008, 5797296695, 16839886311, 48946364931, 142350220626, 414230666047, 1206045545815, 3513292083173, 10239676596500, 29858811813661] The number of three-rowed n-celled Young tableaux Y such that Y[2, 1] = 7, equals M(n - 1) - 4 M(n - 2) + 2 M(n - 3) + 6 M(n - 4) - 3 M(n - 5) - 2 M(n - 6) The first, 30, non-zero terms are [1, 3, 8, 22, 61, 171, 482, 1364, 3870, 11002, 31328, 89332, 255060, 729132, 2086776, 5979072, 17150025, 49244139, 141543312, 407244766, 1172841769, 3380862991, 9754534738, 28168465356, 81411264695, 235482075101, 681664912308, 1974748658398, 5724914188407, 16608533033561, 48215646547186] The number of three-rowed n-celled Young tableaux Y such that Y[2, 2] = 7, equals 5 M(n - 2) - 15 M(n - 3) + 15 M(n - 4) - 15 M(n - 6) The first, 30, non-zero terms are [15, 45, 125, 340, 920, 2495, 6790, 18560, 50955, 140490, 388905, 1080600, 3012945, 8427720, 23643795, 66514560, 187595085, 530337285, 1502577855, 4265900820, 12134288390, 34577523475, 98696218160, 282156159200, 807830305525, 2316087479615, 6649051494195, 19111854482700, 54999067053240, 158449319112545, 456965295980470] The number of three-rowed n-celled Young tableaux Y such that Y[2, 3] = 7, equals 9 M(n - 3) - 18 M(n - 4) + 7 M(n - 5) + 2 M(n - 6) The first, 30, non-zero terms are [25, 59, 152, 390, 1020, 2692, 7177, 19289, 52221, 142279, 389849, 1073603, 2970015, 8249877, 23000829, 64343067, 180549291, 508059441, 1433375646, 4053642672, 11489300382, 32631371514, 92855736675, 264702495751, 755845127779, 2161660102265, 6191277607044, 17757162655942, 50995597813460, 146631037421180, 422108501528667] The number of three-rowed n-celled Young tableaux Y such that Y[3, 1] = 7, equals 5 M(n - 3) - 6 M(n - 4) + M(n - 5) - 4 M(n - 6) The first, 30, non-zero terms are [19, 47, 122, 314, 820, 2160, 5745, 15405, 41615, 113155, 309479, 850847, 2350197, 6519129, 18152415, 50721039, 142173657, 399679437, 1126585644, 3183362220, 9015649482, 25587353246, 72762363511, 207292444467, 591565962705, 1690902753789, 4840462834082, 13876145226682, 39831691007084, 114481127591720, 329423266605995] The number of three-rowed n-celled Young tableaux Y such that Y[3, 2] = 7, equals 16 M(n - 6) The first, 30, non-zero terms are [16, 32, 64, 144, 336, 816, 2032, 5168, 13360, 35008, 92768, 248176, 669360, 1818144, 4969152, 13655472, 37708464, 104582112, 291188544, 813632304, 2280760944, 6412211568, 18076166640, 51083644752, 144694440016, 410717095616, 1168124364832, 3328372451344, 9499884557264, 27158167539376, 77756186822256] The number of three-rowed n-celled Young tableaux Y such that Y[1, 4] = 8, equals 21 M(n - 5) - 42 M(n - 7) The first, 30, non-zero terms are [42, 105, 273, 693, 1785, 4641, 12201, 32382, 86688, 233835, 635019, 1734852, 4764942, 13150179, 36448335, 101418408, 283200246, 793364355, 2229128811, 6280242885, 17737971237, 50215228371, 142461515091, 404971620522, 1153340323800, 3290356466397, 9402272023725, 26908117210653, 77117798241393, 221314766489985, 635941769889321] The number of three-rowed n-celled Young tableaux Y such that Y[1, 5] = 8, equals 14 M(n - 3) - 7 M(n - 4) - 56 M(n - 5) - 7 M(n - 6) + 56 M(n - 7) The first, 30, non-zero terms are [49, 147, 406, 1134, 3136, 8666, 23947, 66283, 183869, 511357, 1425921, 3986801, 11175927, 31407243, 88473441, 249792585, 706766739, 2003781549, 5691835716, 16197005916, 46169277938, 131815302108, 376907038501, 1079250589781, 3094542573955, 8884348948489, 25537641550022, 73491156817246, 211720765212264, 610580009144686, 1762587667898001] The number of three-rowed n-celled Young tableaux Y such that Y[1, 6] = 8, equals 14 M(n - 2) - 41 M(n - 3) - 18 M(n - 4) + 87 M(n - 5) + 17 M(n - 6) - 44 M(n - 7) The first, 30, non-zero terms are [29, 72, 201, 559, 1575, 4450, 12600, 35714, 101325, 287735, 817894, 2327332, 6629805, 18907749, 53986290, 154323864, 441656787, 1265410278, 3629627589, 10422327935, 29958894077, 86204680276, 248293409352, 715836697462, 2065676849025, 5966174351774, 17246479646457, 49895592571391, 144466723603779, 418605160041900, 1213836392553310] The number of three-rowed n-celled Young tableaux Y such that Y[1, 7] = 8, equals 6 M(n - 1) - 30 M(n - 2) + 30 M(n - 3) + 42 M(n - 4) - 54 M(n - 5) - 18 M(n - 6) + 18 M(n - 7) The first, 30, non-zero terms are [6, 18, 48, 132, 366, 1026, 2898, 8232, 23478, 67158, 192528, 552888, 1589958, 4577670, 13193172, 38058822, 109883178, 317507238, 918131496, 2656873404, 7693780674, 22294694022, 64646774826, 187571743974, 544571932806, 1581986686506, 4598358859788, 13373600108208, 38916378055062, 113304840499506, 330057815835066] The number of three-rowed n-celled Young tableaux Y such that Y[1, 8] = 8, equals M(n) - 7 M(n - 1) + 15 M(n - 2) - 4 M(n - 3) - 19 M(n - 4) + 12 M(n - 5) + 6 M(n - 6) - 3 M(n - 7) The first, 30, non-zero terms are [1, 2, 5, 13, 35, 96, 267, 750, 2123, 6045, 17292, 49644, 142935, 412491, 1192635, 3453624, 10014003, 29068420, 84459997, 245609885, 714769229, 2081514358, 6065423840, 17684407602, 51588231825, 150566218296, 439652402517, 1284358731805, 3753613587323, 10974671730410, 32100039337396] The number of three-rowed n-celled Young tableaux Y such that Y[2, 1] = 8, equals M(n - 1) - 5 M(n - 2) + 5 M(n - 3) + 7 M(n - 4) - 9 M(n - 5) - 3 M(n - 6) + 3 M(n - 7) The first, 30, non-zero terms are [1, 3, 8, 22, 61, 171, 483, 1372, 3913, 11193, 32088, 92148, 264993, 762945, 2198862, 6343137, 18313863, 52917873, 153021916, 442812234, 1282296779, 3715782337, 10774462471, 31261957329, 90761988801, 263664447751, 766393143298, 2228933351368, 6486063009177, 18884140083251, 55009635972511] The number of three-rowed n-celled Young tableaux Y such that Y[2, 2] = 8, equals 6 M(n - 2) - 24 M(n - 3) + 33 M(n - 4) - 12 M(n - 5) - 27 M(n - 6) + 24 M(n - 7) The first, 30, non-zero terms are [21, 63, 174, 486, 1350, 3750, 10425, 29031, 81015, 226605, 635331, 1785453, 5029065, 14196411, 40158675, 113826861, 323243523, 919582857, 2620510776, 7479547260, 21380677548, 61205328504, 175446901143, 503570312313, 1447117459815, 4163417087901, 11991501014478, 34574137488534, 99784151957466, 288259651733250, 833487703270515] The number of three-rowed n-celled Young tableaux Y such that Y[2, 3] = 8, equals 14 M(n - 3) - 42 M(n - 4) + 49 M(n - 5) - 7 M(n - 6) - 28 M(n - 7) The first, 30, non-zero terms are [70, 189, 490, 1288, 3402, 9072, 24374, 65961, 179634, 491995, 1354374, 3745427, 10400502, 28988757, 81073818, 227446527, 639901458, 1805022891, 5103876750, 14463960210, 41074653662, 116868522526, 333120779586, 951120346743, 2719898170766, 7789530939435, 22339489549314, 64151017251160, 184446420172938, 530938143804292, 1530021760220422] The number of three-rowed n-celled Young tableaux Y such that Y[2, 4] = 8, equals 14 M(n - 4) - 42 M(n - 5) + 14 M(n - 6) + 28 M(n - 7) The first, 30, non-zero terms are [14, 28, 70, 182, 490, 1330, 3640, 10010, 27650, 76664, 213304, 595378, 1666770, 4679052, 13169184, 37153914, 105057498, 297689700, 845195862, 2404112802, 6850268502, 19551083594, 55886034400, 159980456566, 458592165934, 1316285039384, 3782743167518, 10883524456070, 31348175017642, 90388050966850, 260881668537140] The number of three-rowed n-celled Young tableaux Y such that Y[3, 1] = 8, equals 6 M(n - 3) - 10 M(n - 4) + 4 M(n - 5) - 10 M(n - 6) + 2 M(n - 7) The first, 30, non-zero terms are [34, 96, 254, 680, 1820, 4902, 13272, 36134, 98882, 271894, 750918, 2082290, 5795622, 16185918, 45345474, 127403946, 358912038, 1013591748, 2868998874, 8138043972, 23129641584, 65859719330, 187854304916, 536692645722, 1535643638334, 4400240000744, 12625469249066, 36271959909304, 104331772752540, 300439022950130, 866093040566312] The number of three-rowed n-celled Young tableaux Y such that Y[3, 2] = 8, equals 35 M(n - 6) - 14 M(n - 7) The first, 30, non-zero terms are [56, 112, 259, 609, 1491, 3731, 9527, 24703, 64890, 172298, 461713, 1247071, 3391500, 9279144, 25523337, 70538727, 195778464, 545465592, 1525030689, 4277236299, 12031046979, 33930929403, 95928827085, 271820898377, 771836011646, 2195894589406, 6258705918087, 17868671574089, 51096092504779, 146328262076731, 419638263665271] The number of three-rowed n-celled Young tableaux Y such that Y[1, 4] = 9, equals 42 M(n - 7) - 42 M(n - 9) The first, 30, non-zero terms are [42, 126, 294, 714, 1764, 4452, 11424, 29736, 78330, 208446, 559566, 1513554, 4121166, 11286954, 31072986, 85940694, 238682430, 665385210, 1861256754, 5222627550, 14696270568, 41462939952, 117262512108, 332372967612, 944037808518, 2686503552642, 7658845308786, 21870870505134, 62553212106084, 179172793445604, 513919722770832] The number of three-rowed n-celled Young tableaux Y such that Y[1, 5] = 9, equals 14 M(n - 4) + 28 M(n - 5) - 84 M(n - 6) - 56 M(n - 7) + 42 M(n - 9) The first, 30, non-zero terms are [140, 364, 1022, 2786, 7588, 20650, 56350, 154280, 424004, 1169714, 3238900, 9000054, 25091892, 70172970, 196817712, 553516530, 1560590724, 4410271992, 12490836078, 35449076670, 100797182780, 287124007982, 819257225206, 2341298178772, 6700982594908, 19205626830740, 55117891534034, 158379648718382, 455637859053076, 1312281623031638, 3783522020441450] The number of three-rowed n-celled Young tableaux Y such that Y[1, 6] = 9, equals 28 M(n - 3) - 48 M(n - 4) - 108 M(n - 5) + 112 M(n - 6) + 116 M(n - 7) - 36 M(n - 9) The first, 30, non-zero terms are [92, 276, 764, 2164, 6116, 17292, 48852, 138012, 390040, 1103144, 3123176, 8852760, 25126080, 71410080, 203232240, 579192624, 1652884668, 4723219572, 13514301372, 38716034548, 111048022868, 318884890780, 916728328484, 2638224143052, 7600256771892, 21916549873692, 63259529058196, 182756480033084, 528441168383964, 1529263217211956, 4429095298204972] The number of three-rowed n-celled Young tableaux Y such that Y[1, 7] = 9, equals 20 M(n - 2) - 79 M(n - 3) + 15 M(n - 4) + 186 M(n - 5) - 57 M(n - 6) - 126 M(n - 7) + 21 M(n - 9) The first, 30, non-zero terms are [41, 102, 285, 793, 2235, 6336, 18047, 51549, 147534, 422832, 1213158, 3483906, 10013292, 28802412, 82911060, 238848492, 688582089, 1986594966, 5735618525, 16571614905, 47913586839, 138629857168, 401377929339, 1162900937493, 3371459927507, 9780759767166, 28392234321327, 82469225665535, 239685948023493, 697018814880180, 2028106335920401] The number of three-rowed n-celled Young tableaux Y such that Y[1, 8] = 9, equals 7 M(n - 1) - 42 M(n - 2) + 63 M(n - 3) + 42 M(n - 4) - 126 M(n - 5) + 63 M(n - 7) - 7 M(n - 9) The first, 30, non-zero terms are [7, 21, 56, 154, 427, 1197, 3381, 9611, 27454, 78722, 226394, 652582, 1884512, 5450088, 15781059, 45741843, 132701079, 385276549, 1119369356, 3254247598, 9466375303, 27552229649, 80233701184, 233761734486, 681393082713, 1987111254947, 5797492750616, 16921775533726, 49412176746009, 144344238794567, 421830259758922] The number of three-rowed n-celled Young tableaux Y such that Y[1, 9] = 9, equals M(n) - 8 M(n - 1) + 21 M(n - 2) - 13 M(n - 3) - 25 M(n - 4) + 30 M(n - 5) + 6 M(n - 6) - 12 M(n - 7) + M(n - 9) The first, 30, non-zero terms are [1, 2, 5, 13, 35, 96, 267, 750, 2123, 6046, 17302, 49709, 143275, 414051, 1199187, 3479454, 10111123, 29420490, 85699977, 249876715, 729175029, 2129391033, 6222450290, 18193698327, 53224349337, 155779366096, 456145481717, 1336217082505, 3915789338123, 11479433795315, 33664491472546] The number of three-rowed n-celled Young tableaux Y such that Y[2, 1] = 9, equals M(n - 1) - 6 M(n - 2) + 9 M(n - 3) + 6 M(n - 4) - 18 M(n - 5) + 9 M(n - 7) - M(n - 9) The first, 30, non-zero terms are [1, 3, 8, 22, 61, 171, 483, 1373, 3922, 11246, 32342, 93226, 269216, 778584, 2254437, 6534549, 18957297, 55039507, 159909908, 464892514, 1352339329, 3936032807, 11461957312, 33394533498, 97341868959, 283873036421, 828213250088, 2417396504818, 7058882392287, 20620605542081, 60261465679846] The number of three-rowed n-celled Young tableaux Y such that Y[2, 2] = 9, equals 7 M(n - 2) - 35 M(n - 3) + 63 M(n - 4) - 42 M(n - 5) - 42 M(n - 6) + 84 M(n - 7) - 21 M(n - 9) The first, 30, non-zero terms are [28, 84, 231, 644, 1806, 5082, 14329, 40467, 114450, 324156, 919422, 2611560, 7428582, 21160440, 60359292, 172404645, 493084872, 1412028324, 4048503487, 11621310204, 33396833946, 96078366986, 276692269245, 797629364322, 2301549012364, 6647178254808, 19214814338349, 55590597847036, 160960026601794, 466414707563826, 1352541554856527] The number of three-rowed n-celled Young tableaux Y such that Y[2, 3] = 9, equals 20 M(n - 3) - 80 M(n - 4) + 124 M(n - 5) - 48 M(n - 6) - 44 M(n - 7) - 36 M(n - 9) The first, 30, non-zero terms are [140, 420, 1148, 3092, 8348, 22612, 61564, 168436, 463040, 1278560, 3544864, 9865248, 27549480, 77178072, 216840264, 610875672, 1725218844, 4883538324, 13853280972, 39376216196, 112129661852, 319859783444, 913910411900, 2615224895476, 7494389827684, 21505479302380, 61789459874900, 177746730686108, 511897188174852, 1475814917032268, 4259171896239812] The number of three-rowed n-celled Young tableaux Y such that Y[2, 4] = 9, equals 28 M(n - 4) - 84 M(n - 5) + 70 M(n - 6) - 28 M(n - 7) + 42 M(n - 9) The first, 30, non-zero terms are [98, 224, 574, 1526, 4088, 11060, 30100, 82390, 226618, 626108, 1736798, 4835544, 13508334, 37853004, 106373946, 299717628, 846538854, 2396421384, 6798181530, 19322967342, 55023778684, 156953758892, 448426738592, 1283120835206, 3676726351942, 10549634009488, 30308422881394, 87178472358194, 251042690445176, 723690465781948, 2088341379525380] The number of three-rowed n-celled Young tableaux Y such that Y[3, 1] = 9, equals 7 M(n - 3) - 15 M(n - 4) + 10 M(n - 5) - 21 M(n - 6) + 10 M(n - 7) + M(n - 9) The first, 30, non-zero terms are [69, 186, 517, 1413, 3873, 10624, 29233, 80689, 223468, 620928, 1730792, 4839002, 13567498, 38141796, 107494638, 303659808, 859681689, 2438807370, 6931878417, 19738151893, 56298485321, 160834025456, 460161128865, 1318425646117, 3782516991847, 10865609802994, 31249719112351, 89976670721547, 259346550281823, 748297939453468, 2161178020636359] The number of three-rowed n-celled Young tableaux Y such that Y[3, 2] = 9, equals 64 M(n - 6) - 58 M(n - 7) - 6 M(n - 9) The first, 30, non-zero terms are [134, 338, 810, 2022, 5116, 13180, 34400, 90840, 242230, 651410, 1764674, 4811358, 13192770, 36357702, 100650966, 279771834, 780523218, 2184830070, 6134408382, 17271892146, 48755287512, 137953189584, 391194968500, 1111572474916, 3164491620938, 9024778567854, 25780169182078, 73757542000610, 211328250289532, 606319804249596, 1741826410733040] The number of three-rowed n-celled Young tableaux Y such that Y[3, 3] = 9, equals 42 M(n - 9) The first, 30, non-zero terms are [42, 42, 84, 168, 378, 882, 2142, 5334, 13566, 35070, 91896, 243516, 651462, 1757070, 4772628, 13044024, 35845614, 98984718, 274528044, 764369928, 2135784798, 5986997478, 16832055366, 47449937430, 134094567474, 379822905042, 1078132375992, 3066326457684, 8736977684778, 24937196962818, 71290189790862] The number of three-rowed n-celled Young tableaux Y such that Y[1, 4] = 10, equals 42 M(n - 9) The first, 30, non-zero terms are [42, 84, 168, 378, 882, 2142, 5334, 13566, 35070, 91896, 243516, 651462, 1757070, 4772628, 13044024, 35845614, 98984718, 274528044, 764369928, 2135784798, 5986997478, 16832055366, 47449937430, 134094567474, 379822905042, 1078132375992, 3066326457684, 8736977684778, 24937196962818, 71290189790862, 204109990408422] The number of three-rowed n-celled Young tableaux Y such that Y[1, 5] = 10, equals 84 M(n - 6) - 252 M(n - 8) The first, 30, non-zero terms are [252, 756, 2016, 5376, 14280, 38136, 102396, 276612, 751548, 2053044, 5636484, 15545628, 43055460, 119705292, 333982404, 934831548, 2624401332, 7387775388, 20849401944, 58977889992, 167196802752, 474946185504, 1351697347140, 3853715485116, 11005161113604, 31476435179532, 90158513473056, 258596799039936, 742678686378216, 2135547098740440, 6147791872668516] The number of three-rowed n-celled Young tableaux Y such that Y[1, 6] = 10, equals 42 M(n - 4) - 6 M(n - 5) - 240 M(n - 6) - 72 M(n - 7) + 360 M(n - 8) + 36 M(n - 9) The first, 30, non-zero terms are [324, 852, 2436, 6864, 19338, 54288, 152286, 427176, 1199262, 3371076, 9490590, 26763984, 75608406, 213971796, 606598038, 1722602952, 4899878568, 13959650556, 39831308280, 113816880528, 325681012230, 933155770632, 2677103015514, 7689527991864, 22112283811380, 63656644865004, 183446008881132, 529183788808176, 1527984911342814, 4415984219828688, 12773637011446098] The number of three-rowed n-celled Young tableaux Y such that Y[1, 7] = 10, equals 48 M(n - 3) - 135 M(n - 4) - 132 M(n - 5) + 402 M(n - 6) + 198 M(n - 7) - 315 M(n - 8) - 66 M(n - 9) The first, 30, non-zero terms are [153, 459, 1272, 3606, 10245, 29235, 83538, 238932, 683742, 1957578, 5607408, 16071252, 46090404, 132272748, 379882728, 1091849328, 3140649153, 9041175603, 26048386272, 75107970654, 216739052529, 625934992695, 1809072382818, 5232510844860, 15145513725663, 43870172263893, 127162091345124, 368842018254462, 1070554845403791, 3109243359329265, 9035856995117730] The number of three-rowed n-celled Young tableaux Y such that Y[1, 8] = 10, equals 27 M(n - 2) - 134 M(n - 3) + 102 M(n - 4) + 307 M(n - 5) - 296 M(n - 6) - 255 M(n - 7) + 165 M(n - 8) + 57 M(n - 9) The first, 30, non-zero terms are [55, 137, 383, 1066, 3005, 8520, 24297, 69573, 199843, 575358, 1659378, 4792227, 13854687, 40090638, 116097183, 336430854, 975528657, 2830327845, 8216247187, 23863886264, 69347817455, 201624251968, 586497706652, 1706865019326, 4969782307443, 14476949061945, 42190478520667, 123011533016848, 358812391243961, 1047069455948150, 3056794398887440] The number of three-rowed n-celled Young tableaux Y such that Y[1, 9] = 10, equals 8 M(n - 1) - 56 M(n - 2) + 112 M(n - 3) + 16 M(n - 4) - 232 M(n - 5) + 88 M(n - 6) + 144 M(n - 7) - 48 M(n - 8) - 24 M(n - 9) The first, 30, non-zero terms are [8, 24, 64, 176, 488, 1368, 3864, 10984, 31384, 90048, 259248, 748440, 2165592, 6277656, 18225504, 52980048, 154173272, 449057048, 1308988736, 3818298528, 11144737912, 32546862656, 95096887256, 277986764112, 812958826920, 2378417455128, 6961024526064, 20380544332064, 59691037177896, 174882901927168, 512537162709848] The number of three-rowed n-celled Young tableaux Y such that Y[1, 10] = 10, equals M(n) - 9 M(n - 1) + 28 M(n - 2) - 27 M(n - 3) - 27 M(n - 4) + 59 M(n - 5) - 5 M(n - 6) - 30 M(n - 7) + 6 M(n - 8) + 4 M(n - 9) The first, 30, non-zero terms are [1, 2, 5, 13, 35, 96, 267, 750, 2123, 6046, 17303, 49720, 143352, 414480, 1201266, 3488617, 10148831, 29567846, 86253123, 251887713, 736298794, 2154092458, 6306587420, 18476003823, 54159512731, 158843299826, 466089016747, 1368221296615, 4018054148078, 11804128731650, 34689613957636] The number of three-rowed n-celled Young tableaux Y such that Y[2, 1] = 10, equals M(n - 1) - 7 M(n - 2) + 14 M(n - 3) + 2 M(n - 4) - 29 M(n - 5) + 11 M(n - 6) + 18 M(n - 7) - 6 M(n - 8) - 3 M(n - 9) The first, 30, non-zero terms are [1, 3, 8, 22, 61, 171, 483, 1373, 3923, 11256, 32406, 93555, 270699, 784707, 2278188, 6622506, 19271659, 56132131, 163623592, 477287316, 1393092239, 4068357832, 11887110907, 34748345514, 101619853365, 297302181891, 870128065758, 2547568041508, 7461379647237, 21860362740896, 64067145338731] The number of three-rowed n-celled Young tableaux Y such that Y[2, 2] = 10, equals 8 M(n - 2) - 48 M(n - 3) + 108 M(n - 4) - 104 M(n - 5) - 40 M(n - 6) + 192 M(n - 7) - 60 M(n - 8) - 64 M(n - 9) The first, 30, non-zero terms are [36, 108, 296, 824, 2308, 6516, 18480, 52584, 149968, 428456, 1225840, 3511544, 10070424, 28910112, 83077128, 238960368, 687969396, 1982435180, 5717473768, 16503414248, 47675717124, 137836431524, 398807979832, 1154746127040, 3345969956460, 9701984883396, 28150919728728, 81735112874856, 237465067768764, 690330335679308, 2008037412389336] The number of three-rowed n-celled Young tableaux Y such that Y[2, 3] = 10, equals 27 M(n - 3) - 135 M(n - 4) + 267 M(n - 5) - 207 M(n - 6) - 18 M(n - 7) + 90 M(n - 8) - 144 M(n - 9) The first, 30, non-zero terms are [252, 756, 2148, 5979, 16530, 45690, 126462, 350898, 976338, 2724372, 7623522, 21390588, 60173946, 169687692, 479606922, 1358479827, 3855637512, 10963790532, 31231787328, 89116528671, 254684339286, 728935831650, 2089214302722, 5995834772325, 17228925401412, 49565537085912, 142753867832916, 411583240032633, 1187862756573294, 3431565610772010, 9922382113381770] The number of three-rowed n-celled Young tableaux Y such that Y[2, 4] = 10, equals 48 M(n - 4) - 192 M(n - 5) + 306 M(n - 6) - 180 M(n - 7) - 90 M(n - 8) + 108 M(n - 9) The first, 30, non-zero terms are [378, 966, 2568, 6828, 18360, 49656, 135186, 370050, 1018098, 2813742, 7808490, 21750678, 60793830, 170451594, 479276802, 1351191654, 3818607822, 10816144434, 30700761156, 87311627616, 248762284140, 709962099684, 2029445602614, 5809903784382, 16656015765822, 47813292917538, 137426084120016, 395460346253868, 1139259322851528, 3285510500981640, 9484630015860726] The number of three-rowed n-celled Young tableaux Y such that Y[2, 5] = 10, equals 42 M(n - 5) - 168 M(n - 6) + 126 M(n - 7) + 126 M(n - 8) - 84 M(n - 9) The first, 30, non-zero terms are [42, 84, 210, 546, 1470, 4032, 11172, 31164, 87318, 245490, 692076, 1955688, 5538078, 15712830, 44660700, 127152144, 362582766, 1035473544, 2961322938, 8480406690, 24316618578, 69810159804, 200649496404, 577348562196, 1663016914650, 4795049614992, 13839102272394, 39978155819490, 115590017020926, 334490920966020, 968722540310112] The number of three-rowed n-celled Young tableaux Y such that Y[3, 1] = 10, equals 8 M(n - 3) - 21 M(n - 4) + 20 M(n - 5) - 40 M(n - 6) + 30 M(n - 7) - 3 M(n - 8) + 6 M(n - 9) The first, 30, non-zero terms are [125, 367, 1024, 2870, 7999, 22305, 62232, 173910, 486908, 1366068, 3840810, 10821738, 30554418, 86441766, 245024874, 695817990, 1979436105, 5640396771, 16097682068, 46011600754, 131700740695, 377480398313, 1083318607372, 3112760653938, 8954407227555, 25787251539225, 74340836954252, 214527287226014, 619655390989405, 1791478821689899, 5183811653492780] The number of three-rowed n-celled Young tableaux Y such that Y[3, 2] = 10, equals 105 M(n - 6) - 153 M(n - 7) + 27 M(n - 8) - 36 M(n - 9) The first, 30, non-zero terms are [351, 864, 2241, 5775, 15105, 39849, 106134, 284943, 770577, 2097270, 5740884, 15795207, 43658577, 121175964, 337595940, 943766595, 2646609813, 7443205704, 20988192177, 59326181865, 168071970087, 477145964841, 1357222632906, 3867567584217, 11039777582055, 31562527241586, 90371177089089, 259117216100631, 743935912385265, 2138530379741505, 6154692026318664] The number of three-rowed n-celled Young tableaux Y such that Y[3, 3] = 10, equals 168 M(n - 9) The first, 30, non-zero terms are [168, 336, 672, 1512, 3528, 8568, 21336, 54264, 140280, 367584, 974064, 2605848, 7028280, 19090512, 52176096, 143382456, 395938872, 1098112176, 3057479712, 8543139192, 23947989912, 67328221464, 189799749720, 536378269896, 1519291620168, 4312529503968, 12265305830736, 34947910739112, 99748787851272, 285160759163448, 816439961633688] The number of three-rowed n-celled Young tableaux Y such that Y[1, 5] = 11, equals 252 M(n - 8) - 42 M(n - 9) - 462 M(n - 10) The first, 30, non-zero terms are [462, 1176, 3066, 7812, 20160, 52500, 138180, 367080, 983430, 2654400, 7212282, 19712616, 54163746, 149530752, 414578430, 1153881792, 3222859626, 9030530376, 25378130862, 71511701940, 202010596956, 571964858388, 1622893551048, 4613931108696, 13141774414650, 37496083515072, 107156613057606, 306697187249796, 879060586068672, 2522957397262260, 7250201316518580] The number of three-rowed n-celled Young tableaux Y such that Y[1, 6] = 11, equals 42 M(n - 5) + 120 M(n - 6) - 330 M(n - 7) - 360 M(n - 8) + 528 M(n - 10) The first, 30, non-zero terms are [780, 2340, 6528, 18312, 50874, 141174, 391638, 1087914, 3027798, 8445858, 23615778, 66193566, 185980374, 523749330, 1478234394, 4181009094, 11849233248, 33645312576, 95706054396, 272704409484, 778291864854, 2224602824298, 6367758922698, 18252050948694, 52383760473180, 150526411960260, 433044809730000, 1247188010249688, 3595725549624942, 10377070888522674, 29976176101918074] The number of three-rowed n-celled Young tableaux Y such that Y[1, 7] = 11, equals 90 M(n - 4) - 135 M(n - 5) - 495 M(n - 6) + 405 M(n - 7) + 810 M(n - 8) + 45 M(n - 9) - 495 M(n - 10) The first, 30, non-zero terms are [630, 1665, 4770, 13545, 38700, 110475, 315315, 899550, 2566035, 7320915, 20895030, 59672880, 170541045, 487799955, 1396510740, 4001788530, 11478379860, 32955405795, 94708890450, 272438135145, 784423817190, 2260634516685, 6520734393435, 18825147547020, 54393215009040, 157291079476905, 455202909762840, 1318366629016245, 3821083018354170, 11082684634686675, 32166302765630175] The number of three-rowed n-celled Young tableaux Y such that Y[1, 8] = 11, equals 75 M(n - 3) - 290 M(n - 4) - 50 M(n - 5) + 950 M(n - 6) - 80 M(n - 7) - 990 M(n - 8) - 105 M(n - 9) + 330 M(n - 10) The first, 30, non-zero terms are [235, 705, 1955, 5545, 15760, 45060, 129255, 371695, 1070605, 3087255, 8910105, 25732755, 74360355, 214994625, 621921420, 1799957970, 5212048605, 15099945255, 43768825285, 126934132815, 368312726690, 1069251394570, 3105753556060, 9025614930240, 26242621431285, 76340617231735, 222187255627405, 646985097558055, 1884852660545220, 5493696389626160, 16019611381966960] The number of three-rowed n-celled Young tableaux Y such that Y[1, 9] = 11, equals 35 M(n - 2) - 209 M(n - 3) + 273 M(n - 4) + 400 M(n - 5) - 809 M(n - 6) - 264 M(n - 7) + 642 M(n - 8) + 111 M(n - 9) - 143 M(n - 10) The first, 30, non-zero terms are [71, 177, 495, 1378, 3885, 11016, 31417, 90000, 258778, 746260, 2157122, 6247024, 18118620, 52614711, 152941680, 444949224, 1295411543, 3773789645, 10999930927, 32079058210, 93595530209, 273197638483, 797767909785, 2330481314652, 6810474898525, 19909774300091, 58224770055717, 170332575163970, 498462602238963, 1459181067611235, 4272918257806291] The number of three-rowed n-celled Young tableaux Y such that Y[1, 10] = 11, equals 9 M(n - 1) - 72 M(n - 2) + 180 M(n - 3) - 54 M(n - 4) - 360 M(n - 5) + 306 M(n - 6) + 225 M(n - 7) - 216 M(n - 8) - 54 M(n - 9) + 36 M(n - 10) The first, 30, non-zero terms are [9, 27, 72, 198, 549, 1539, 4347, 12357, 35307, 101313, 291753, 842679, 2440044, 7080264, 20581371, 59917491, 174658383, 509684013, 1488750462, 4352086296, 12731596236, 37268423388, 109154470383, 319859475615, 937720208439, 2750213370789, 8069082119892, 23682875456862, 69532381871928, 204208377679224, 599910479630739] The number of three-rowed n-celled Young tableaux Y such that Y[1, 11] = 11, equals M(n) - 10 M(n - 1) + 36 M(n - 2) - 47 M(n - 3) - 21 M(n - 4) + 99 M(n - 5) - 39 M(n - 6) - 55 M(n - 7) + 30 M(n - 8) + 10 M(n - 9) - 4 M(n - 10) The first, 30, non-zero terms are [1, 2, 5, 13, 35, 96, 267, 750, 2123, 6046, 17303, 49721, 143364, 414570, 1201798, 3491332, 10161359, 29621566, 86470995, 252733650, 739470654, 2165651488, 6347729336, 18619570996, 54652165555, 160509753326, 471656616627, 1386623541760, 4078308523658, 11999794215500, 35320374980536] The number of three-rowed n-celled Young tableaux Y such that Y[2, 1] = 11, equals M(n - 1) - 8 M(n - 2) + 20 M(n - 3) - 6 M(n - 4) - 40 M(n - 5) + 34 M(n - 6) + 25 M(n - 7) - 24 M(n - 8) - 6 M(n - 9) + 4 M(n - 10) The first, 30, non-zero terms are [1, 3, 8, 22, 61, 171, 483, 1373, 3923, 11257, 32417, 93631, 271116, 786696, 2286819, 6657499, 19406487, 56631557, 165416718, 483565144, 1414621804, 4140935932, 12128274487, 35539941735, 104191134271, 305579263421, 896564679988, 2631430606318, 7725820207992, 22689819742136, 66656719958971] The number of three-rowed n-celled Young tableaux Y such that Y[2, 2] = 11, equals 9 M(n - 2) - 63 M(n - 3) + 171 M(n - 4) - 216 M(n - 5) + 9 M(n - 6) + 360 M(n - 7) - 270 M(n - 8) - 135 M(n - 9) + 99 M(n - 10) The first, 30, non-zero terms are [45, 135, 369, 1026, 2871, 8100, 22995, 65592, 187758, 538884, 1549782, 4464144, 12875652, 37177029, 107446500, 310797756, 899703837, 2606355963, 7555484385, 21916495314, 63613385235, 184750109685, 536872473639, 1560990000096, 4541136938991, 13217763375405, 38492311789755, 112151913431274, 326926628840601, 953452927792701, 2781942654294405] The number of three-rowed n-celled Young tableaux Y such that Y[2, 3] = 11, equals 35 M(n - 3) - 210 M(n - 4) + 510 M(n - 5) - 570 M(n - 6) + 145 M(n - 7) + 240 M(n - 8) - 370 M(n - 9) + 220 M(n - 10) The first, 30, non-zero terms are [420, 1260, 3555, 10055, 28300, 79560, 223680, 629400, 1773370, 5004350, 14145730, 40054390, 113610930, 322790550, 918608625, 2618324325, 7474321800, 21367174280, 61167538315, 175333880375, 503216546720, 1445981232260, 4159726353415, 11979486083535, 34535134584600, 99658215421920, 287855595413385, 832199816513205, 2407990356269880, 6973324549140300, 20210068325002835] The number of three-rowed n-celled Young tableaux Y such that Y[2, 4] = 11, equals 75 M(n - 4) - 375 M(n - 5) + 765 M(n - 6) - 645 M(n - 7) + 90 M(n - 8) + 180 M(n - 9) - 330 M(n - 10) The first, 30, non-zero terms are [1050, 2940, 7950, 21555, 58470, 159420, 436590, 1201260, 3319440, 9209550, 25645740, 71658570, 200852820, 564595470, 1591291080, 4496013945, 12731900130, 36130665960, 102733582350, 292647829275, 835069638870, 2386698098760, 6831669101910, 19582656342555, 56207734027110, 161535315664200, 464788882386690, 1338853679117025, 3860771405412930, 11144357355339120, 32199866566498170] The number of three-rowed n-celled Young tableaux Y such that Y[2, 5] = 11, equals 90 M(n - 5) - 360 M(n - 6) + 468 M(n - 7) - 324 M(n - 8) + 18 M(n - 9) + 396 M(n - 10) The first, 30, non-zero terms are [378, 846, 2160, 5724, 15570, 42750, 118260, 328680, 917100, 2567268, 7207488, 20287656, 57243240, 161876664, 458718768, 1302425568, 3704685066, 10555874910, 30125784384, 86108063724, 246475568394, 706471388838, 2027563691220, 5826188417112, 16760909807478, 48271250426418, 139166272982376, 401617025209260, 1160119020176694, 3354179307195690, 9706116799153620] The number of three-rowed n-celled Young tableaux Y such that Y[3, 1] = 11, equals 9 M(n - 3) - 28 M(n - 4) + 35 M(n - 5) - 71 M(n - 6) + 71 M(n - 7) - 18 M(n - 8) + 21 M(n - 9) - 3 M(n - 10) The first, 30, non-zero terms are [251, 702, 2020, 5723, 16205, 45774, 129273, 365180, 1032452, 2922222, 8281680, 23503170, 66796950, 190113330, 541861047, 1546570953, 4420194591, 12649786050, 36247207004, 103990645971, 298690305637, 858883366154, 2472368003330, 7124235015291, 20548999773837, 59327003983166, 171437656952132, 495833355284345, 1435242951103743, 4157770618359310, 12053885603269904] The number of three-rowed n-celled Young tableaux Y such that Y[3, 2] = 11, equals 160 M(n - 6) - 325 M(n - 7) + 138 M(n - 8) - 123 M(n - 9) + 22 M(n - 10) The first, 30, non-zero terms are [763, 2129, 5624, 15058, 40340, 108780, 294875, 803755, 2201895, 6060565, 16753403, 46495849, 129510189, 361945983, 1014652935, 2852468913, 8040078609, 22717026099, 64330767978, 182555032080, 519056973274, 1478517839182, 4218691398857, 12056534806389, 34507919983865, 98907262669003, 283867402167324, 815733895259794, 2346915366142748, 6759816355672340, 19491067361119825] The number of three-rowed n-celled Young tableaux Y such that Y[3, 3] = 11, equals 450 M(n - 9) - 198 M(n - 10) The first, 30, non-zero terms are [702, 1404, 3258, 7668, 18792, 47052, 120204, 311796, 819270, 2175876, 5831946, 15754572, 42851970, 117257868, 322566894, 891564084, 2474729658, 6895474164, 19279950318, 54077701788, 152119033668, 429041068596, 1213034946480, 3437371021644, 9760824619002, 27770873702652, 79154936179254, 225995644087668, 646262390648808, 1850810431076172, 5307873394092732] The number of three-rowed n-celled Young tableaux Y such that Y[1, 5] = 12, equals 462 M(n - 10) - 462 M(n - 12) The first, 30, non-zero terms are [462, 1386, 3234, 7854, 19404, 48972, 125664, 327096, 861630, 2292906, 6155226, 16649094, 45332826, 124156494, 341802846, 945347634, 2625506730, 7319237310, 20473824294, 57448903050, 161658976248, 456092339472, 1289887633188, 3656102643732, 10384415893698, 29551539079062, 84247298396646, 240579575556474, 688085333166924, 1970900727901644, 5653116950479152] The number of three-rowed n-celled Young tableaux Y such that Y[1, 6] = 12, equals 330 M(n - 7) - 1320 M(n - 9) + 330 M(n - 12) The first, 30, non-zero terms are [1980, 5280, 14850, 40590, 110880, 302610, 827970, 2272380, 6258780, 17300250, 47988600, 133560570, 372900660, 1044235170, 2932314660, 8255630790, 23299283700, 65904935580, 186815165130, 530598482370, 1509820243800, 4303680099510, 12287525586930, 35136342357600, 100618459067940, 288530801089080, 828451917331830, 2381621627439450, 6854575645080240, 19749896734349430, 56964136626113310] The number of three-rowed n-celled Young tableaux Y such that Y[1, 7] = 12, equals 132 M(n - 5) + 33 M(n - 6) - 990 M(n - 7) - 495 M(n - 8) + 1980 M(n - 9) + 495 M(n - 10) - 297 M(n - 12) The first, 30, non-zero terms are [1815, 5445, 15345, 43923, 125037, 355509, 1009041, 2862189, 8118000, 23034330, 65405538, 185892300, 528903342, 1506591900, 4296733452, 12269072574, 35076340035, 100401210261, 287723151969, 825483432807, 2370961651509, 6817220515281, 19621871092989, 56533687112529, 163039092277281, 470628041944995, 1359721571595255, 3931817518726845, 11378722544251623, 32956163441611047, 95523354556216059] The number of three-rowed n-celled Young tableaux Y such that Y[1, 8] = 12, equals 165 M(n - 4) - 440 M(n - 5) - 715 M(n - 6) + 1705 M(n - 7) + 1485 M(n - 8) - 1815 M(n - 9) - 990 M(n - 10) + 220 M(n - 12) The first, 30, non-zero terms are [1100, 2915, 8360, 23760, 68090, 195525, 562540, 1619750, 4666145, 13446235, 38758060, 111750650, 322318315, 930013755, 2684632005, 7753356105, 22403679870, 64771691325, 187368152620, 542321214160, 1570619404200, 4551349447535, 13196658420605, 38286068357625, 111139381141030, 322806444295775, 938122235300690, 2727816825307310, 7936055682431360, 23100580044842465, 67276676463396675] The number of three-rowed n-celled Young tableaux Y such that Y[1, 9] = 12, equals 110 M(n - 3) - 539 M(n - 4) + 264 M(n - 5) + 1749 M(n - 6) - 1254 M(n - 7) - 2178 M(n - 8) + 990 M(n - 9) + 979 M(n - 10) - 121 M(n - 12) The first, 30, non-zero terms are [341, 1023, 2838, 8052, 22891, 65461, 187913, 541233, 1562792, 4521286, 13099812, 37998466, 110321486, 320533554, 931868817, 2710618779, 7888422377, 22966951007, 66895474558, 194922810984, 568192944539, 1656885582847, 4833369107512, 14104769921308, 41175494393099, 120244919940761, 351275713195858, 1026552894506348, 3000990683593617, 8775993112805061, 25672864289976486] The number of three-rowed n-celled Young tableaux Y such that Y[1, 10] = 12, equals 44 M(n - 2) - 307 M(n - 3) + 564 M(n - 4) + 373 M(n - 5) - 1685 M(n - 6) + 195 M(n - 7) + 1614 M(n - 8) - 258 M(n - 9) - 540 M(n - 10) + 45 M(n - 12) The first, 30, non-zero terms are [89, 222, 621, 1729, 4875, 13824, 39427, 112950, 324819, 937086, 2710674, 7858521, 22824879, 66396627, 193394223, 563912294, 1645803819, 4807120230, 14050305929, 41090664615, 120233989161, 351976827433, 1030823531178, 3020107772475, 8851478118345, 25951000656924, 76107756304197, 223271475348941, 655180924676919, 1923127184847915, 5646370011543514] The number of three-rowed n-celled Young tableaux Y such that Y[1, 11] = 12, equals 10 M(n - 1) - 90 M(n - 2) + 270 M(n - 3) - 190 M(n - 4) - 480 M(n - 5) + 720 M(n - 6) + 200 M(n - 7) - 600 M(n - 8) + 160 M(n - 10) - 10 M(n - 12) The first, 30, non-zero terms are [10, 30, 80, 220, 610, 1710, 4830, 13730, 39230, 112570, 324180, 936430, 2712050, 7872150, 22894300, 66693770, 194567230, 568326330, 1661877360, 4864220790, 14249174290, 41771963870, 122535535330, 359658144040, 1056189504890, 3103096506510, 9120758148980, 26818370655730, 78883393935270, 232101801623210, 683126034669610] The number of three-rowed n-celled Young tableaux Y such that Y[1, 12] = 12, equals M(n) - 11 M(n - 1) + 45 M(n - 2) - 74 M(n - 3) - 2 M(n - 4) + 147 M(n - 5) - 111 M(n - 6) - 75 M(n - 7) + 90 M(n - 8) + 10 M(n - 9) - 20 M(n - 10) + M(n - 12) The first, 30, non-zero terms are [1, 2, 5, 13, 35, 96, 267, 750, 2123, 6046, 17303, 49721, 143365, 414583, 1201902, 3491982, 10164843, 29638362, 86545914, 253048575, 740734059, 2170532949, 6366017463, 18686351151, 54890802837, 161346965976, 474547726862, 1396471458085, 4111454821973, 12110194818215, 35684699401711] The number of three-rowed n-celled Young tableaux Y such that Y[2, 1] = 12, equals M(n - 1) - 9 M(n - 2) + 27 M(n - 3) - 19 M(n - 4) - 48 M(n - 5) + 72 M(n - 6) + 20 M(n - 7) - 60 M(n - 8) + 16 M(n - 10) - M(n - 12) The first, 30, non-zero terms are [1, 3, 8, 22, 61, 171, 483, 1373, 3923, 11257, 32418, 93643, 271205, 787215, 2289430, 6669377, 19456723, 56832633, 166187736, 486422079, 1424917429, 4177196387, 12253553533, 35965814404, 105618950489, 310309650651, 912075814898, 2681837065573, 7888339393527, 23210180162321, 68312603466961] The number of three-rowed n-celled Young tableaux Y such that Y[2, 2] = 12, equals 10 M(n - 2) - 80 M(n - 3) + 255 M(n - 4) - 400 M(n - 5) + 155 M(n - 6) + 570 M(n - 7) - 750 M(n - 8) - 120 M(n - 9) + 405 M(n - 10) - 45 M(n - 12) The first, 30, non-zero terms are [55, 165, 450, 1250, 3495, 9855, 27965, 79785, 228600, 657210, 1894500, 5472810, 15836880, 45891810, 133138485, 386631265, 1123715475, 3268413765, 9512720170, 27703393230, 80723704545, 235338346985, 686427458160, 2003070222810, 5847743739105, 17079043271355, 49901689350990, 145860503315950, 426506617757775, 1247598309406995, 3650735931792830] The number of three-rowed n-celled Young tableaux Y such that Y[2, 3] = 12, equals 44 M(n - 3) - 308 M(n - 4) + 891 M(n - 5) - 1287 M(n - 6) + 715 M(n - 7) + 330 M(n - 8) - 990 M(n - 9) + 1100 M(n - 10) - 341 M(n - 12) The first, 30, non-zero terms are [660, 1980, 5555, 15664, 44330, 125730, 357093, 1015399, 2890426, 8236844, 23498442, 67112540, 191892118, 549285396, 1574059608, 4515629481, 12968178080, 37281306524, 107285657875, 309041541600, 891051189270, 2571490549518, 7427620348049, 21472528803198, 62125801610268, 179888381746920, 521271458543385, 1511619163022760, 4386587612729454, 12738140878679150, 37014290319146635] The number of three-rowed n-celled Young tableaux Y such that Y[2, 4] = 12, equals 110 M(n - 4) - 660 M(n - 5) + 1650 M(n - 6) - 1980 M(n - 7) + 990 M(n - 8) + 440 M(n - 9) - 1210 M(n - 10) - 220 M(n - 12) The first, 30, non-zero terms are [2310, 6930, 19470, 53790, 148390, 409530, 1133220, 3144900, 8755120, 24448160, 68471260, 192299140, 541476980, 1528422060, 4324121010, 12259664010, 34827733710, 99124779050, 282617194750, 807100142750, 2308478244710, 6612325395610, 18966031591210, 54470228403690, 156628820244930, 450905535341910, 1299495183393330, 3748994097020370, 10826381860375290, 31293890738768790, 90536565453886530] The number of three-rowed n-celled Young tableaux Y such that Y[2, 5] = 12, equals 165 M(n - 5) - 825 M(n - 6) + 1683 M(n - 7) - 1584 M(n - 8) + 792 M(n - 10) + 297 M(n - 12) The first, 30, non-zero terms are [1848, 4554, 11979, 32406, 88308, 242550, 669405, 1855755, 5164038, 14419548, 40389228, 113454594, 319538934, 902164824, 2552892606, 7239286593, 20569047444, 58550711274, 166954917855, 476836224282, 1363955370624, 3907090150062, 11207111226597, 32187583997316, 92556633428292, 266454782665038, 767910664417149, 2215365445447374, 6397430159789856, 18491441846088618, 53496085767432495] The number of three-rowed n-celled Young tableaux Y such that Y[2, 6] = 12, equals 132 M(n - 6) - 660 M(n - 7) + 792 M(n - 8) + 396 M(n - 9) - 792 M(n - 10) + 132 M(n - 12) The first, 30, non-zero terms are [132, 264, 660, 1716, 4620, 12672, 35244, 98868, 279048, 790944, 2249016, 6410712, 18309984, 52384464, 150091920, 430611984, 1236915108, 3556975752, 10239504660, 29505907860, 85103921628, 245686833216, 709886627868, 2052841749156, 5941102420044, 17207207110632, 49873863035964, 144658216342860, 419864834956116, 1219443227848560, 3543963671168052] The number of three-rowed n-celled Young tableaux Y such that Y[3, 1] = 12, equals 10 M(n - 3) - 36 M(n - 4) + 56 M(n - 5) - 119 M(n - 6) + 146 M(n - 7) - 63 M(n - 8) + 60 M(n - 9) - 21 M(n - 10) - M(n - 12) The first, 30, non-zero terms are [461, 1373, 3933, 11317, 32371, 92469, 263777, 752167, 2144890, 6118886, 17466890, 49900510, 142687160, 408397242, 1170074862, 3355703868, 9633689989, 27684475709, 79635557397, 229295345829, 660830370167, 1906248366011, 5503658012071, 15903524937771, 45993121593359, 133118250642551, 385580863522743, 1117672958260103, 3242082231890477, 9410925132655341, 27335626111425049] The number of three-rowed n-celled Young tableaux Y such that Y[3, 2] = 12, equals 231 M(n - 6) - 605 M(n - 7) + 429 M(n - 8) - 363 M(n - 9) + 154 M(n - 10) + 11 M(n - 12) The first, 30, non-zero terms are [1804, 4851, 13442, 36718, 100672, 276353, 761101, 2102815, 5829329, 16212361, 45230460, 126561028, 355122911, 999061437, 2817530562, 7964161062, 22560198342, 64035140169, 182100839910, 518768344842, 1480323657670, 4230765948959, 12109360215601, 34707801754043, 99610095903802, 286232616304459, 823466393976776, 2371686216004072, 6837993397410318, 19735015785229217, 57011501675941813] The number of three-rowed n-celled Young tableaux Y such that Y[3, 3] = 12, equals 990 M(n - 9) - 990 M(n - 10) The first, 30, non-zero terms are [1980, 4950, 11880, 29700, 75240, 194040, 506880, 1339470, 3573900, 9615870, 26060760, 71081010, 194968620, 537466050, 1488278880, 4137806970, 11546272980, 32326207650, 90778584600, 255633507360, 721707220080, 2042337708180, 5792167956960, 16460151815250, 46864574782740, 133665350352930, 381862311553800, 1092606259518180, 3130752443128200, 8983069593612840, 25808206475082240] The number of three-rowed n-celled Young tableaux Y such that Y[3, 4] = 12, equals 462 M(n - 12) The first, 30, non-zero terms are [462, 462, 924, 1848, 4158, 9702, 23562, 58674, 149226, 385770, 1010856, 2678676, 7166082, 19327770, 52498908, 143484264, 394301754, 1088831898, 3019808484, 8408069208, 23493632778, 65856972258, 185152609026, 521949311730, 1475040242214, 4178051955462, 11859456135912, 33729591034524, 96106754532558, 274309166590998, 784192087699482] The number of three-rowed n-celled Young tableaux Y such that Y[1, 5] = 13, equals 462 M(n - 12) The first, 30, non-zero terms are [462, 924, 1848, 4158, 9702, 23562, 58674, 149226, 385770, 1010856, 2678676, 7166082, 19327770, 52498908, 143484264, 394301754, 1088831898, 3019808484, 8408069208, 23493632778, 65856972258, 185152609026, 521949311730, 1475040242214, 4178051955462, 11859456135912, 33729591034524, 96106754532558, 274309166590998, 784192087699482, 2245209894492642] The number of three-rowed n-celled Young tableaux Y such that Y[1, 6] = 13, equals 1320 M(n - 9) - 528 M(n - 10) - 3432 M(n - 11) + 528 M(n - 12) The first, 30, non-zero terms are [3432, 10296, 27456, 73392, 195360, 522720, 1405800, 3802920, 10344840, 28288920, 77735592, 214568376, 594691416, 1654433352, 4618528200, 12934010232, 36326959416, 102304178856, 288827435952, 817307409600, 2317732922736, 6585796261968, 18748315567608, 53465398836696, 152718722707320, 436895172978792, 1251664844834976, 3590776937397936, 10314426625712352, 29663846344793760, 85409735377500600] The number of three-rowed n-celled Young tableaux Y such that Y[1, 7] = 13, equals 132 M(n - 6) + 495 M(n - 7) - 1287 M(n - 8) - 1980 M(n - 9) + 4290 M(n - 11) - 132 M(n - 12) The first, 30, non-zero terms are [5610, 15180, 43758, 123882, 350229, 986304, 2774805, 7804830, 21967209, 61895130, 174636363, 493488138, 1396742985, 3959720226, 11243890251, 31978580634, 91090448064, 259857165744, 742369410024, 2123750541990, 6083584047519, 17448698364900, 50106000006579, 144050968147350, 414591086736222, 1194482297450640, 3444888504056670, 9944594931091458, 28734081050130027, 83097626486798484, 240516442893746679] The number of three-rowed n-celled Young tableaux Y such that Y[1, 8] = 13, equals 297 M(n - 5) - 374 M(n - 6) - 2145 M(n - 7) + 1342 M(n - 8) + 4730 M(n - 9) + 660 M(n - 10) - 4290 M(n - 11) - 220 M(n - 12) The first, 30, non-zero terms are [3630, 10890, 30745, 88385, 253825, 730147, 2099559, 6035865, 17346901, 49848623, 143253187, 411761669, 1183958325, 3405829911, 9802610268, 28230574086, 81353478492, 234597856020, 676973334103, 1954892768939, 5649101789507, 16335778864889, 47271720267310, 136886266256476, 396652376660298, 1150127639808510, 3337039875765475, 9688339983250995, 28145113257378975, 81811662766067245, 237946289148526442] The number of three-rowed n-celled Young tableaux Y such that Y[1, 9] = 13, equals 275 M(n - 4) - 1034 M(n - 5) - 605 M(n - 6) + 4389 M(n - 7) + 726 M(n - 8) - 6226 M(n - 9) - 1694 M(n - 10) + 3146 M(n - 11) + 407 M(n - 12) The first, 30, non-zero terms are [1782, 4730, 13574, 38599, 110660, 318142, 917565, 2651759, 7675118, 22237501, 64478370, 187063151, 542951310, 1576539030, 4579371951, 13306341906, 38677817618, 112464971432, 327135234602, 951908358533, 2770917670488, 8068916832955, 23505630537356, 68500687985620, 199703206129746, 582428168449402, 1699286383850110, 4959716440263189, 14481447620683604, 42299031947243225, 123597695898983482] The number of three-rowed n-celled Young tableaux Y such that Y[1, 10] = 13, equals 154 M(n - 3) - 912 M(n - 4) + 994 M(n - 5) + 2632 M(n - 6) - 4134 M(n - 7) - 3120 M(n - 8) + 4598 M(n - 9) + 2064 M(n - 10) - 1612 M(n - 11) - 344 M(n - 12) The first, 30, non-zero terms are [474, 1422, 3946, 11198, 31840, 91064, 261436, 753212, 2176280, 6303080, 18291064, 53163880, 154723866, 450773862, 1314439302, 3835661434, 11199784706, 32719880006, 95635241746, 279644786102, 818015205546, 2393696912222, 7006808642890, 20516678869118, 60092963609654, 176061839354434, 515974349475382, 1512547956545346, 4435123909836498, 13008154661555590, 38162514779945354] The number of three-rowed n-celled Young tableaux Y such that Y[1, 11] = 13, equals 54 M(n - 2) - 431 M(n - 3) + 1017 M(n - 4) + 82 M(n - 5) - 2943 M(n - 6) + 1701 M(n - 7) + 2975 M(n - 8) - 1895 M(n - 9) - 1326 M(n - 10) + 546 M(n - 11) + 166 M(n - 12) The first, 30, non-zero terms are [109, 272, 761, 2119, 5975, 16944, 48327, 138450, 398159, 1148740, 3323453, 9638014, 28007190, 81528528, 237684060, 693823591, 2027583323, 5930935544, 17363096931, 50867950065, 149119858216, 437389591828, 1283559452132, 3768386596701, 11067967655755, 32519018197064, 95576426206987, 280993838286751, 826353819272624, 2430807418507700, 7152264763878244] The number of three-rowed n-celled Young tableaux Y such that Y[1, 12] = 13, equals 11 M(n - 1) - 110 M(n - 2) + 385 M(n - 3) - 418 M(n - 4) - 539 M(n - 5) + 1386 M(n - 6) - 132 M(n - 7) - 1254 M(n - 8) + 385 M(n - 9) + 440 M(n - 10) - 110 M(n - 11) - 44 M(n - 12) The first, 30, non-zero terms are [11, 33, 88, 242, 671, 1881, 5313, 15103, 43153, 123827, 356598, 1030084, 2983398, 8660498, 25190726, 73400162, 214200085, 625934639, 1831285797, 5363412351, 15722879634, 46130005658, 135442819336, 397937779052, 1169852646545, 3440946484811, 10125879775143, 29810959497473, 87799140741642, 258678708737746, 762384578021396] The number of three-rowed n-celled Young tableaux Y such that Y[1, 13] = 13, equals M(n) - 12 M(n - 1) + 55 M(n - 2) - 109 M(n - 3) + 36 M(n - 4) + 196 M(n - 5) - 237 M(n - 6) - 63 M(n - 7) + 204 M(n - 8) - 25 M(n - 9) - 60 M(n - 10) + 10 M(n - 11) + 5 M(n - 12) The first, 30, non-zero terms are [1, 2, 5, 13, 35, 96, 267, 750, 2123, 6046, 17303, 49721, 143365, 414584, 1201916, 3492101, 10165627, 29642765, 86568048, 253151118, 741180255, 2172380585, 6373367575, 18714641105, 54996725381, 161734410061, 475936933072, 1401367594930, 4128454750793, 12168453152825, 35882062233811] The number of three-rowed n-celled Young tableaux Y such that Y[2, 1] = 13, equals M(n - 1) - 10 M(n - 2) + 35 M(n - 3) - 38 M(n - 4) - 49 M(n - 5) + 126 M(n - 6) - 12 M(n - 7) - 114 M(n - 8) + 35 M(n - 9) + 40 M(n - 10) - 10 M(n - 11) - 4 M(n - 12) The first, 30, non-zero terms are [1, 3, 8, 22, 61, 171, 483, 1373, 3923, 11257, 32418, 93644, 271218, 787318, 2290066, 6672742, 19472735, 56903149, 166480527, 487582941, 1429352694, 4193636878, 12312983576, 36176161732, 106350240595, 312813316801, 920534525013, 2710087227043, 7981740067422, 23516246248886, 69307688911036] The number of three-rowed n-celled Young tableaux Y such that Y[2, 2] = 13, equals 11 M(n - 2) - 99 M(n - 3) + 363 M(n - 4) - 682 M(n - 5) + 473 M(n - 6) + 759 M(n - 7) - 1650 M(n - 8) + 275 M(n - 9) + 1056 M(n - 10) - 286 M(n - 11) - 176 M(n - 12) The first, 30, non-zero terms are [66, 198, 539, 1496, 4180, 11781, 33418, 95315, 273086, 785345, 2265472, 6551776, 18987540, 55122837, 160258835, 466486834, 1359262982, 3964151356, 11569870279, 33790716905, 98747067034, 288723746927, 844599777373, 2471793637544, 7236890805650, 21196231184986, 62104392378633, 182026581851499, 533690126013066, 1565234464099955, 4591983659524481] The number of three-rowed n-celled Young tableaux Y such that Y[2, 3] = 13, equals 54 M(n - 3) - 432 M(n - 4) + 1454 M(n - 5) - 2568 M(n - 6) + 2142 M(n - 7) + 16 M(n - 8) - 2230 M(n - 9) + 3360 M(n - 10) - 780 M(n - 11) - 1336 M(n - 12) The first, 30, non-zero terms are [990, 2970, 8294, 23330, 65896, 187200, 533748, 1525956, 4371160, 12540392, 36021832, 103582712, 298146006, 858935274, 2476626858, 7146837846, 20639930166, 59653103330, 172535714878, 499387114538, 1446437129550, 4192349131178, 12159140115478, 35288063532114, 102476540129874, 297772777680150, 865768594744410, 2518648164513534, 7331194341786390, 21350910723460946, 62213638731011030] The number of three-rowed n-celled Young tableaux Y such that Y[2, 4] = 13, equals 154 M(n - 4) - 1078 M(n - 5) + 3201 M(n - 6) - 4917 M(n - 7) + 3729 M(n - 8) - 55 M(n - 9) - 2750 M(n - 10) + 1430 M(n - 11) - 1210 M(n - 12) The first, 30, non-zero terms are [4620, 13860, 39930, 112915, 316954, 888228, 2489256, 6984890, 19632734, 55289080, 156014298, 441118436, 1249656870, 3546833334, 10084860918, 28723749219, 81944020048, 234132711340, 669947598782, 1919642943207, 5507693865718, 15821933611726, 45505334015020, 131024453655909, 377664096464556, 1089682280771040, 3147124650487950, 9097629496836705, 26322336344091150, 76222873726022034, 220899528643023208] The number of three-rowed n-celled Young tableaux Y such that Y[2, 5] = 13, equals 275 M(n - 5) - 1650 M(n - 6) + 4158 M(n - 7) - 5082 M(n - 8) + 2497 M(n - 9) + 528 M(n - 10) - 2002 M(n - 11) + 1276 M(n - 12) The first, 30, non-zero terms are [6468, 17292, 47091, 128183, 351340, 966504, 2670624, 7407576, 20621722, 57600158, 161386786, 453475990, 1277587410, 3608216502, 10213721793, 28973160117, 82350372456, 234496506728, 668893128475, 1911083671079, 5468427618368, 15669888442340, 44962885523479, 129179734674111, 371583395823288, 1070069628933888, 3084862944950457, 8902336540660101, 25715465396033688, 74350954024849260, 215159955992009699] The number of three-rowed n-celled Young tableaux Y such that Y[2, 6] = 13, equals 297 M(n - 6) - 1485 M(n - 7) + 2640 M(n - 8) - 2541 M(n - 9) + 792 M(n - 10) + 2574 M(n - 11) - 1419 M(n - 12) The first, 30, non-zero terms are [1452, 3201, 8151, 21549, 58542, 161667, 450681, 1264494, 3563406, 10075296, 28562226, 81149376, 231000066, 658697490, 1881231264, 5380595649, 15410243376, 44192130741, 126883889847, 364726417473, 1049549154258, 3023366748699, 8717876504997, 25161886787103, 72689275826772, 210172205394327, 608194076050341, 1761392490392271, 5105086181801034, 14807089931291097, 42977731742363847] The number of three-rowed n-celled Young tableaux Y such that Y[3, 1] = 13, equals 11 M(n - 3) - 45 M(n - 4) + 84 M(n - 5) - 190 M(n - 6) + 273 M(n - 7) - 169 M(n - 8) + 154 M(n - 9) - 84 M(n - 10) + 6 M(n - 11) - 8 M(n - 12) The first, 30, non-zero terms are [923, 2626, 7681, 22198, 64124, 184706, 531378, 1527264, 4387772, 12604600, 36214508, 104084068, 299290563, 861093948, 2479053423, 7141978118, 20590204327, 59404465802, 171512798677, 495555866102, 1432863229215, 4145993261146, 12004903706597, 34784730315002, 100858270076501, 292630750495798, 849584164167271, 2468102428781786, 7174350306184843, 20866853365181254, 60726576742393621] The number of three-rowed n-celled Young tableaux Y such that Y[3, 2] = 13, equals 320 M(n - 6) - 1029 M(n - 7) + 1044 M(n - 8) - 967 M(n - 9) + 604 M(n - 10) - 52 M(n - 11) + 80 M(n - 12) The first, 30, non-zero terms are [3774, 11002, 30640, 85780, 239043, 666773, 1861401, 5205455, 14585273, 40952491, 115230435, 324913469, 918031809, 2598995403, 7371822423, 20947292937, 59624743548, 169993910532, 485414097194, 1388126912698, 3975127923965, 11398492189747, 32725712305575, 94069543002353, 270708380314110, 779871826285154, 2249009363470232, 6492094752126060, 18757906315659625, 54246460120944343, 157010506833405583] The number of three-rowed n-celled Young tableaux Y such that Y[3, 3] = 13, equals 1925 M(n - 9) - 3102 M(n - 10) + 572 M(n - 11) - 143 M(n - 12) The first, 30, non-zero terms are [5918, 14509, 37609, 96998, 253990, 670780, 1788325, 4805405, 13005410, 35420825, 97016458, 267069979, 738545874, 2050742463, 5715578220, 15983796543, 44837748714, 126136375959, 355771533513, 1005886646160, 2850336464514, 8093622004622, 23026450568737, 65628451886899, 187364628931150, 535756313792183, 1534226117166479, 4399625039494454, 12633139543548238, 36319870136987140, 104540670070402775] The number of three-rowed n-celled Young tableaux Y such that Y[3, 4] = 13, equals 2112 M(n - 12) The first, 30, non-zero terms are [2112, 4224, 8448, 19008, 44352, 107712, 268224, 682176, 1763520, 4621056, 12245376, 32759232, 88355520, 239995008, 655928064, 1802522304, 4977517248, 13804838784, 38436887808, 107399464128, 301060444608, 846411926976, 2386053996480, 6743041107264, 19099666082112, 54214656621312, 154192416157824, 439345163577408, 1253984761558848, 3584878115197632, 10263816660537792] The number of three-rowed n-celled Young tableaux Y such that Y[1, 6] = 14, equals 3432 M(n - 11) - 858 M(n - 12) - 6006 M(n - 13) The first, 30, non-zero terms are [6006, 15444, 40326, 102960, 265980, 693264, 1825824, 4852848, 13006422, 35117940, 95446494, 260938392, 717124122, 1980142164, 5490895410, 15284823840, 42696912258, 119651633244, 336287290482, 947696349600, 2677342259592, 7581123445896, 21512255881116, 61164054514992, 174223296583050, 497122291982316, 1420756581380994, 4066607818889616, 11656326173391900, 33455837339461680, 96145727983637904] The number of three-rowed n-celled Young tableaux Y such that Y[1, 7] = 14, equals 1287 M(n - 8) - 6435 M(n - 10) + 5148 M(n - 13) The first, 30, non-zero terms are [12870, 38610, 108108, 303732, 845559, 2351349, 6536673, 18194319, 50732253, 141761763, 397025343, 1114494381, 3135630069, 8841593475, 24983754939, 70739964009, 200682224028, 570357406476, 1623819248766, 4630636942074, 13225687838649, 37829837574243, 108356683778343, 310778108793009, 892460742248790, 2565926682655050, 7385672256353820, 21281509615462668, 61384534406975637, 177230565979368759, 512178983571568599] The number of three-rowed n-celled Young tableaux Y such that Y[1, 8] = 14, equals 429 M(n - 6) + 286 M(n - 7) - 4004 M(n - 8) - 2860 M(n - 9) + 10010 M(n - 10) + 4290 M(n - 11) - 5005 M(n - 13) The first, 30, non-zero terms are [12870, 35035, 101530, 291005, 836550, 2399254, 6873867, 19673797, 56281654, 160985825, 460558384, 1318122377, 3774624282, 10816587639, 31019941095, 89032608522, 255757699626, 735337938621, 2116049829322, 6094579128495, 17568551937794, 50686950471872, 146357996245032, 422949042191104, 1223211428520510, 3540357720347305, 10254511177042990, 29723163118212135, 86213932114843390, 250237575357162250, 726792973905106394] The number of three-rowed n-celled Young tableaux Y such that Y[1, 9] = 14, equals 572 M(n - 5) - 1430 M(n - 6) - 3432 M(n - 7) + 6864 M(n - 8) + 9152 M(n - 9) - 9152 M(n - 10) - 9152 M(n - 11) - 286 M(n - 12) + 4004 M(n - 13) The first, 30, non-zero terms are [6578, 19734, 55770, 160446, 461604, 1332760, 3853278, 11151998, 32291974, 93534298, 270977850, 785184686, 2275564434, 6596273970, 19125660840, 55470075804, 160931372030, 467063038442, 1356045134158, 3938633131290, 11444492563504, 33268427598120, 96751178011488, 281495002723792, 819363490579062, 2386020204688978, 6951243819099294, 20260048578997370, 59075326317615036, 172328524241306004, 502910809365066960] The number of three-rowed n-celled Young tableaux Y such that Y[1, 10] = 14, equals 429 M(n - 4) - 2067 M(n - 5) + 390 M(n - 6) + 8775 M(n - 7) - 4290 M(n - 8) - 14664 M(n - 9) + 4173 M(n - 10) + 9984 M(n - 11) + 663 M(n - 12) - 2457 M(n - 13) The first, 30, non-zero terms are [2730, 7254, 20826, 59241, 169884, 488514, 1409616, 4077918, 11822499, 34332558, 99833019, 290595240, 846556074, 2467780614, 7197649056, 21002551557, 61309149252, 179032809930, 522976153596, 1528145859981, 4466587954959, 13059047971398, 38191731658821, 111724361620131, 326923487393334, 956893015463190, 2801554826324514, 8204513866243851, 24033889065557241, 70422614713877634, 206403221075690379] The number of three-rowed n-celled Young tableaux Y such that Y[1, 11] = 14, equals 208 M(n - 3) - 1443 M(n - 4) + 2392 M(n - 5) + 3172 M(n - 6) - 9646 M(n - 7) - 1768 M(n - 8) + 12714 M(n - 9) + 533 M(n - 10) - 6552 M(n - 11) - 702 M(n - 12) + 1092 M(n - 13) The first, 30, non-zero terms are [637, 1911, 5304, 15054, 42809, 122447, 351559, 1012921, 2927015, 8479653, 24619101, 71609187, 208615212, 608555064, 1777224631, 5195167679, 15198840059, 44496367825, 130346875750, 382036559736, 1120237234428, 3286196518332, 9643559327051, 28309142713371, 83128432942659, 244172176153737, 717395135266836, 2108287109881654, 6197331093563064, 18221273663800584, 53585594367879327] The number of three-rowed n-celled Young tableaux Y such that Y[1, 12] = 14, equals 65 M(n - 2) - 584 M(n - 3) + 1680 M(n - 4) - 679 M(n - 5) - 4467 M(n - 6) + 5049 M(n - 7) + 3999 M(n - 8) - 6149 M(n - 9) - 1680 M(n - 10) + 2630 M(n - 11) + 400 M(n - 12) - 329 M(n - 13) The first, 30, non-zero terms are [131, 327, 915, 2548, 7185, 20376, 58117, 166500, 478833, 1381511, 3996993, 11592031, 33689799, 98092280, 286068068, 835440892, 2442860849, 7150773201, 20951819460, 61440632175, 180306818549, 529485795553, 1555787823941, 4573755140006, 13452355730255, 39582785599331, 116514361679652, 343085694710345, 1010561705555043, 2977480736329125, 8775084926213641] The number of three-rowed n-celled Young tableaux Y such that Y[1, 13] = 14, equals 12 M(n - 1) - 132 M(n - 2) + 528 M(n - 3) - 768 M(n - 4) - 456 M(n - 5) + 2328 M(n - 6) - 1080 M(n - 7) - 2088 M(n - 8) + 1548 M(n - 9) + 780 M(n - 10) - 600 M(n - 11) - 120 M(n - 12) + 60 M(n - 13) The first, 30, non-zero terms are [12, 36, 96, 264, 732, 2052, 5796, 16476, 47076, 135084, 389016, 1123728, 3254628, 9447984, 27482208, 80082132, 233724036, 683092176, 1998930960, 5856004908, 17172747540, 50404391700, 148063461780, 435254782092, 1280337605052, 3768455682432, 11097778097496, 32697975362556, 96382635534684, 284218862234532, 838432513971732] The number of three-rowed n-celled Young tableaux Y such that Y[1, 14] = 14, equals M(n) - 13 M(n - 1) + 66 M(n - 2) - 153 M(n - 3) + 100 M(n - 4) + 234 M(n - 5) - 431 M(n - 6) + 27 M(n - 7) + 378 M(n - 8) - 154 M(n - 9) - 125 M(n - 10) + 60 M(n - 11) + 15 M(n - 12) - 5 M(n - 13) The first, 30, non-zero terms are [1, 2, 5, 13, 35, 96, 267, 750, 2123, 6046, 17303, 49721, 143365, 414584, 1201917, 3492116, 10165762, 29643700, 86573538, 253179846, 741318290, 2173001600, 6376019290, 18725493540, 55039609640, 161898959536, 476552753472, 1403623470580, 4136566858268, 12197157047600, 35982199092436] The number of three-rowed n-celled Young tableaux Y such that Y[2, 1] = 14, equals M(n - 1) - 11 M(n - 2) + 44 M(n - 3) - 64 M(n - 4) - 38 M(n - 5) + 194 M(n - 6) - 90 M(n - 7) - 174 M(n - 8) + 129 M(n - 9) + 65 M(n - 10) - 50 M(n - 11) - 10 M(n - 12) + 5 M(n - 13) The first, 30, non-zero terms are [1, 3, 8, 22, 61, 171, 483, 1373, 3923, 11257, 32418, 93644, 271219, 787332, 2290184, 6673511, 19477003, 56924348, 166577580, 488000409, 1431062295, 4200365975, 12338621815, 36271231841, 106694800421, 314037973536, 924814841458, 2724831280213, 8031886294557, 23684905186211, 69869376164311] The number of three-rowed n-celled Young tableaux Y such that Y[2, 2] = 14, equals 12 M(n - 2) - 120 M(n - 3) + 498 M(n - 4) - 1092 M(n - 5) + 1068 M(n - 6) + 792 M(n - 7) - 3108 M(n - 8) + 1584 M(n - 9) + 2010 M(n - 10) - 1464 M(n - 11) - 444 M(n - 12) + 252 M(n - 13) The first, 30, non-zero terms are [78, 234, 636, 1764, 4926, 13878, 39354, 112218, 321450, 924342, 2666598, 7713882, 22366476, 64979904, 189099174, 551089638, 1608005586, 4696898118, 13731915780, 40178723124, 117642286212, 344665364868, 1010343843114, 2963149407606, 8694211071786, 25520072963598, 74936658106704, 220117842962376, 646776750895524, 1901003977092876, 5588994011599458] The number of three-rowed n-celled Young tableaux Y such that Y[2, 3] = 14, equals 65 M(n - 3) - 585 M(n - 4) + 2249 M(n - 5) - 4693 M(n - 6) + 5161 M(n - 7) - 1586 M(n - 8) - 4056 M(n - 9) + 8515 M(n - 10) - 4680 M(n - 11) - 3510 M(n - 12) + 2184 M(n - 13) The first, 30, non-zero terms are [1430, 4290, 11934, 33501, 94471, 268060, 764465, 2188862, 6286423, 18097014, 52191867, 150740889, 435890637, 1261714974, 3655310165, 10598015116, 30749025112, 89273306798, 259344881120, 753851206914, 2192472942075, 6379903860720, 18574506007369, 54104896947036, 157676020850166, 459725536694190, 1341003820899210, 3913403415668864, 11425296689348451, 33370587687273876, 97507748849937465] The number of three-rowed n-celled Young tableaux Y such that Y[2, 4] = 14, equals 208 M(n - 4) - 1664 M(n - 5) + 5733 M(n - 6) - 10686 M(n - 7) + 10842 M(n - 8) - 3718 M(n - 9) - 5265 M(n - 10) + 6760 M(n - 11) - 4030 M(n - 12) + 1820 M(n - 13) The first, 30, non-zero terms are [8580, 25740, 73645, 210353, 597948, 1696656, 4811768, 13649688, 38749542, 110116786, 313296126, 892496618, 2545795278, 7271217018, 20794589127, 59544354883, 170711184384, 490000705584, 1408075864325, 4050708324897, 11665208607416, 33627307036388, 97031262998553, 280242452235369, 810106573743168, 2343799572982920, 6786622763421735, 19666517140143315, 57033235751205168, 165516914108124748, 480682409891411741] The number of three-rowed n-celled Young tableaux Y such that Y[2, 5] = 14, equals 429 M(n - 5) - 3003 M(n - 6) + 9009 M(n - 7) - 14157 M(n - 8) + 11583 M(n - 9) - 1716 M(n - 10) - 6006 M(n - 11) + 3432 M(n - 12) - 3003 M(n - 13) The first, 30, non-zero terms are [18018, 51480, 142857, 395538, 1093092, 3028311, 8410545, 23429406, 65460252, 183419808, 515348262, 1451690526, 4099167072, 11601075915, 32901696828, 93496740480, 266181596538, 759123705912, 2168471594289, 6203817963486, 17774173517100, 50992809487620, 146482107757557, 421294600259676, 1213066994568858, 3496675264870788, 10089587449018935, 29141830838508102, 84249026287033608, 243780153604990002, 705991199517729411] The number of three-rowed n-celled Young tableaux Y such that Y[2, 6] = 14, equals 572 M(n - 6) - 3432 M(n - 7) + 8580 M(n - 8) - 10868 M(n - 9) + 4004 M(n - 10) + 3432 M(n - 11) - 1716 M(n - 12) + 4004 M(n - 13) The first, 30, non-zero terms are [8580, 20592, 53768, 144716, 397540, 1101672, 3073356, 8610888, 24210472, 68264768, 192961912, 546651248, 1551771936, 4413222528, 12572965548, 35877735036, 102535676100, 293460342032, 841034128792, 2413436749724, 6934053601332, 19945298038952, 57434254583992, 165559990673220, 477718607725788, 1379752352652960, 3988627957248792, 11540422340875428, 33417900504178476, 96845717942247560, 280873616125408592] The number of three-rowed n-celled Young tableaux Y such that Y[2, 7] = 14, equals 429 M(n - 7) - 2574 M(n - 8) + 4290 M(n - 9) + 429 M(n - 10) - 5148 M(n - 11) + 1287 M(n - 12) + 1287 M(n - 13) The first, 30, non-zero terms are [429, 858, 2145, 5577, 15015, 41184, 114543, 321750, 910338, 2589444, 7395102, 21184020, 60828768, 175001112, 504262044, 1454952213, 4202810469, 12152654514, 35172119697, 101879740677, 295333089051, 856741317432, 2487040282155, 7224311227419, 20997990555399, 61068244648554, 177704955734163, 517393539154635, 1507202303681217, 4392821259898500, 12809430268060569] The number of three-rowed n-celled Young tableaux Y such that Y[3, 1] = 14, equals 12 M(n - 3) - 55 M(n - 4) + 120 M(n - 5) - 291 M(n - 6) + 476 M(n - 7) - 385 M(n - 8) + 364 M(n - 9) - 257 M(n - 10) + 48 M(n - 11) - 36 M(n - 12) + 4 M(n - 13) The first, 30, non-zero terms are [1715, 5133, 14893, 43459, 126234, 366218, 1060650, 3068810, 8872652, 25642564, 74095736, 214106884, 618779889, 1788785739, 5172895923, 14965458053, 43315910911, 125435554049, 363429114841, 1053540442435, 3055757841903, 8867976112489, 25749445869381, 74807936884967, 217451044004597, 632420857852875, 1840254961564603, 5357630510898833, 15605798138448279, 45479216574289849, 132601638703979293] The number of three-rowed n-celled Young tableaux Y such that Y[3, 2] = 14, equals 429 M(n - 6) - 1638 M(n - 7) + 2184 M(n - 8) - 2327 M(n - 9) + 1833 M(n - 10) - 416 M(n - 11) + 338 M(n - 12) - 52 M(n - 13) The first, 30, non-zero terms are [8515, 23829, 68406, 193622, 548015, 1548014, 4373148, 12359399, 34962837, 99019102, 280803744, 797427137, 2267762445, 6458369034, 18418686897, 52600643082, 150418967673, 430698969753, 1234764675378, 3544155456816, 10184475649309, 29298234544670, 84372554177717, 243219682262258, 701803609822887, 2026908946266425, 5859193862023944, 16951593364586296, 49083673458137487, 142233887963753820, 412472459170965479] The number of three-rowed n-celled Young tableaux Y such that Y[3, 3] = 14, equals 3432 M(n - 9) - 7722 M(n - 10) + 3432 M(n - 11) - 858 M(n - 12) The first, 30, non-zero terms are [14586, 40326, 106392, 284856, 763620, 2060916, 5591586, 15254382, 41823210, 115201086, 318670638, 884959218, 2466378486, 6896443554, 19342191726, 54400004802, 153395658702, 433576274274, 1228236326460, 3486545450388, 9916196233944, 28253770260144, 80637934918398, 230509446900018, 659907455207286, 1891840374495786, 5430732729512376, 15608925960619992, 44915781955320276, 129392542269936660, 373146095252079006] The number of three-rowed n-celled Young tableaux Y such that Y[3, 4] = 14, equals 6435 M(n - 12) - 3003 M(n - 13) The first, 30, non-zero terms are [9867, 19734, 45903, 108108, 265122, 664092, 1697124, 4403256, 11572275, 30739566, 82401891, 222628692, 605604285, 1657287918, 4559412429, 12602911464, 34984210833, 97483637394, 272580292413, 764584929108, 2150841020328, 6066516311856, 17152532847450, 48606533519544, 138027944182557, 392718553097442, 1119387453550509, 3196040915912988, 9139665924401298, 26175317817529452, 75068654431857372] The number of three-rowed n-celled Young tableaux Y such that Y[1, 6] = 15, equals 6006 M(n - 13) - 6006 M(n - 15) The first, 30, non-zero terms are [6006, 18018, 42042, 102102, 252252, 636636, 1633632, 4252248, 11201190, 29807778, 80017938, 216438222, 589326738, 1614034422, 4443436998, 12289519242, 34131587490, 95150085030, 266159715822, 746835739650, 2101566691224, 5929200413136, 16768539231444, 47529334368516, 134997406618074, 384170008027806, 1095214879156398, 3127534482234162, 8945109331170012, 25621709462721372, 73490520356228976] The number of three-rowed n-celled Young tableaux Y such that Y[1, 7] = 15, equals 6435 M(n - 10) - 4290 M(n - 11) - 21021 M(n - 12) + 7722 M(n - 13) + 2574 M(n - 15) The first, 30, non-zero terms are [30459, 82368, 231660, 634062, 1734876, 4742595, 12996555, 35720685, 98513415, 272627355, 757035279, 2108979873, 5893372485, 16516226727, 46412428491, 130754987649, 369241997553, 1045023240690, 2963745744102, 8421684690276, 23974385866002, 68365681484529, 195265941329745, 558562353191271, 1600058984408313, 4589705368709604, 13182145503277608, 37906204955657610, 109126460664107172, 314499564950394681, 907312161689352705] The number of three-rowed n-celled Young tableaux Y such that Y[1, 8] = 15, equals 429 M(n - 7) + 2002 M(n - 8) - 5005 M(n - 9) - 10010 M(n - 10) + 28600 M(n - 12) - 2145 M(n - 13) - 2431 M(n - 15) The first, 30, non-zero terms are [35035, 105105, 298870, 859144, 2453451, 6993701, 19896305, 56558359, 160740294, 456957930, 1299842258, 3700547422, 10545440048, 30083322984, 85915396281, 245646181209, 703143783771, 2014970932689, 5780632217074, 16601697559876, 47729536491351, 137361617635385, 395705764202034, 1141020082075016, 3293163586554413, 9513007036525135, 27503791005323490, 79583618206807000, 230461192350804129, 667885567617070231, 1936975319085150916] The number of three-rowed n-celled Young tableaux Y such that Y[1, 9] = 15, equals 1001 M(n - 6) - 1001 M(n - 7) - 9009 M(n - 8) + 4004 M(n - 9) + 25025 M(n - 10) + 6006 M(n - 11) - 30030 M(n - 12) - 4004 M(n - 13) + 2002 M(n - 15) The first, 30, non-zero terms are [26026, 71071, 206206, 592592, 1712711, 4950946, 14319305, 41407366, 119714595, 346041696, 1000144145, 2890643756, 8355462115, 24156357225, 69857446659, 202089245358, 584852103836, 1693320872243, 4904970706636, 14215019002184, 41217269402309, 119573824533163, 347072751714689, 1007936446833316, 2928690278824310, 8514120805351161, 24764514838989926, 72067664695442416, 209830276793554417, 611236164839751451, 1781389765373859641] The number of three-rowed n-celled Young tableaux Y such that Y[1, 10] = 15, equals 1001 M(n - 5) - 3640 M(n - 6) - 3822 M(n - 7) + 19110 M(n - 8) + 8463 M(n - 9) - 34398 M(n - 10) - 16562 M(n - 11) + 23296 M(n - 12) + 7917 M(n - 13) - 1365 M(n - 15) The first, 30, non-zero terms are [11102, 33306, 94185, 271089, 780234, 2254434, 6528158, 18940194, 55025607, 160022317, 465703329, 1356034407, 3950121630, 11510515926, 33550846875, 97819552103, 285270238344, 832136901960, 2427961365739, 7085978016663, 20685773653545, 60403301427651, 176428750977024, 515466363400170, 1506453389810278, 4403896840596162, 12877924702714071, 37668883511870975, 110216892830980299, 322583536071875169, 944418757591555974] The number of three-rowed n-celled Young tableaux Y such that Y[1, 11] = 15, equals 637 M(n - 4) - 3731 M(n - 5) + 3185 M(n - 6) + 14560 M(n - 7) - 18382 M(n - 8) - 24479 M(n - 9) + 26572 M(n - 10) + 22386 M(n - 11) - 12649 M(n - 12) - 7462 M(n - 13) + 728 M(n - 15) The first, 30, non-zero terms are [4004, 10647, 30576, 86996, 249522, 717626, 2070978, 5992441, 17380363, 50509550, 147032158, 428600536, 1250811653, 3653797056, 10681707673, 31248187971, 91464479834, 267848099155, 784703711064, 2299759867223, 6742210386548, 19772075130444, 57999249802221, 170178605011233, 499452311572578, 1466172063453279, 4305009101101982, 12643245644236571, 37139510292779076, 109120167680606082, 320673896821232591] The number of three-rowed n-celled Young tableaux Y such that Y[1, 12] = 15, equals 273 M(n - 3) - 2170 M(n - 4) + 4788 M(n - 5) + 2562 M(n - 6) - 18557 M(n - 7) + 5880 M(n - 8) + 26124 M(n - 9) - 10640 M(n - 10) - 17220 M(n - 11) + 4368 M(n - 12) + 4319 M(n - 13) - 287 M(n - 15) The first, 30, non-zero terms are [833, 2499, 6937, 19691, 56000, 160188, 459942, 1325254, 3829693, 11095343, 32216856, 93728054, 273144403, 797181021, 2329595996, 6815392402, 19958356691, 58496245353, 171574911144, 503572712370, 1478833039502, 4345059244162, 12772247100977, 37558928023253, 110488161856891, 325132305672993, 957048044801044, 2817902466945062, 8299038136873596, 24447333749052340, 72032863364999663] The number of three-rowed n-celled Young tableaux Y such that Y[1, 13] = 15, equals 77 M(n - 2) - 769 M(n - 3) + 2607 M(n - 4) - 2188 M(n - 5) - 5926 M(n - 6) + 11163 M(n - 7) + 2919 M(n - 8) - 14433 M(n - 9) + 1080 M(n - 10) + 7790 M(n - 11) - 760 M(n - 12) - 1560 M(n - 13) + 78 M(n - 15) The first, 30, non-zero terms are [155, 387, 1083, 3016, 8505, 24120, 68797, 197100, 566841, 1635443, 4731705, 13722996, 39884066, 116134110, 338716926, 989350548, 2893539645, 8472526200, 24833976699, 72858705930, 213932131632, 628626146502, 1848406848786, 5438287167858, 16008857008861, 47148662291850, 138921426462027, 409489002600458, 1207460282684616, 3561618368469270, 10508815742404570] The number of three-rowed n-celled Young tableaux Y such that Y[1, 14] = 15, equals 13 M(n - 1) - 156 M(n - 2) + 702 M(n - 3) - 1274 M(n - 4) - 117 M(n - 5) + 3510 M(n - 6) - 3055 M(n - 7) - 2730 M(n - 8) + 4095 M(n - 9) + 650 M(n - 10) - 1950 M(n - 11) + 325 M(n - 13) - 13 M(n - 15) The first, 30, non-zero terms are [13, 39, 104, 286, 793, 2223, 6279, 17849, 50999, 146341, 421434, 1217372, 3525847, 10235329, 29772587, 86757385, 253212986, 740085918, 2165867652, 6345711840, 18611404838, 54636850138, 160532107346, 472034864990, 1388966893912, 4089669572248, 12048658284884, 35515782805204, 104740978301076, 309034276402748, 912169386115468] The number of three-rowed n-celled Young tableaux Y such that Y[1, 15] = 15, equals M(n) - 14 M(n - 1) + 78 M(n - 2) - 207 M(n - 3) + 198 M(n - 4) + 243 M(n - 5) - 701 M(n - 6) + 262 M(n - 7) + 588 M(n - 8) - 469 M(n - 9) - 175 M(n - 10) + 210 M(n - 11) + 15 M(n - 12) - 30 M(n - 13) + M(n - 15) The first, 30, non-zero terms are [1, 2, 5, 13, 35, 96, 267, 750, 2123, 6046, 17303, 49721, 143365, 414584, 1201917, 3492117, 10165778, 29643852, 86574642, 253186610, 741355074, 2173184664, 6376869898, 18729235410, 55055352312, 161962786376, 476803602512, 1404583565560, 4140158716748, 12210331676840, 36029694584836] The number of three-rowed n-celled Young tableaux Y such that Y[2, 1] = 15, equals M(n - 1) - 12 M(n - 2) + 54 M(n - 3) - 98 M(n - 4) - 9 M(n - 5) + 270 M(n - 6) - 235 M(n - 7) - 210 M(n - 8) + 315 M(n - 9) + 50 M(n - 10) - 150 M(n - 11) + 25 M(n - 13) - M(n - 15) The first, 30, non-zero terms are [1, 3, 8, 22, 61, 171, 483, 1373, 3923, 11257, 32418, 93644, 271219, 787333, 2290199, 6673645, 19477922, 56929686, 166605204, 488131680, 1431646526, 4202834626, 12348623642, 36310374230, 106843607224, 314589967096, 926819868068, 2731983292708, 8056998330852, 23771867415596, 70166875855036] The number of three-rowed n-celled Young tableaux Y such that Y[2, 2] = 15, equals 13 M(n - 2) - 143 M(n - 3) + 663 M(n - 4) - 1664 M(n - 5) + 2080 M(n - 6) + 429 M(n - 7) - 5187 M(n - 8) + 4641 M(n - 9) + 2730 M(n - 10) - 4550 M(n - 11) - 260 M(n - 12) + 1248 M(n - 13) - 78 M(n - 15) The first, 30, non-zero terms are [91, 273, 741, 2054, 5733, 16146, 45773, 130494, 373737, 1074541, 3099603, 8966256, 25999714, 75550254, 219933870, 641259294, 1872284973, 5473069290, 16015684191, 46909388682, 137507369532, 403369623306, 1184012855142, 3477408499032, 10218224463317, 30039565559796, 88346941470723, 259927901183626, 765004349129124, 2252229133757538, 6632672933512886] The number of three-rowed n-celled Young tableaux Y such that Y[2, 3] = 15, equals 77 M(n - 3) - 770 M(n - 4) + 3332 M(n - 5) - 8022 M(n - 6) + 10892 M(n - 7) - 6230 M(n - 8) - 5719 M(n - 9) + 18830 M(n - 10) - 16590 M(n - 11) - 5488 M(n - 12) + 10976 M(n - 13) - 1288 M(n - 15) The first, 30, non-zero terms are [2002, 6006, 16653, 46669, 131425, 372547, 1061613, 3039071, 8732227, 25166057, 72703414, 210446656, 610125047, 1771183169, 5147333429, 14972817363, 43588504204, 126982839732, 370160169596, 1079642752580, 3150618403163, 9198576255533, 26868372560253, 78514035813507, 229524740624634, 671244038293542, 1963783722189436, 5747295499871128, 16826162683689639, 49278089552346585, 144366292644753697] The number of three-rowed n-celled Young tableaux Y such that Y[2, 4] = 15, equals 273 M(n - 4) - 2457 M(n - 5) + 9646 M(n - 6) - 21112 M(n - 7) + 26754 M(n - 8) - 16107 M(n - 9) - 6370 M(n - 10) + 19110 M(n - 11) - 15015 M(n - 12) + 11830 M(n - 13) - 5278 M(n - 15) The first, 30, non-zero terms are [15015, 45045, 128128, 364637, 1039675, 2968056, 8480472, 24250226, 69397874, 198759652, 569738442, 1634561383, 4693679354, 13490129289, 38806946724, 111735359736, 321998847079, 928738578313, 2681018477228, 7745760419670, 22396308763101, 64807733354637, 187674355797838, 543876701951310, 1577259656769069, 4577234064474435, 13292002929048480, 38623851110223288, 112302469915692147, 326725239295257985, 951101956336346906] The number of three-rowed n-celled Young tableaux Y such that Y[2, 5] = 15, equals 637 M(n - 5) - 5096 M(n - 6) + 17745 M(n - 7) - 33852 M(n - 8) + 36582 M(n - 9) - 16380 M(n - 10) - 8372 M(n - 11) + 14560 M(n - 12) - 16107 M(n - 13) - 1365 M(n - 15) The first, 30, non-zero terms are [42042, 126126, 360360, 1013012, 2837835, 7940205, 22240127, 62393877, 175397586, 494109070, 1394912610, 3946148934, 11185841121, 31768195095, 90386452611, 257607957061, 735393605310, 2102550588138, 6020085209684, 17260533258696, 49552915407996, 142435369998740, 409894762224402, 1180883181282714, 3405630284705922, 9831542253966342, 28409167545565716, 82165328251166160, 237844833988194342, 689059770909884010, 1997846921538229580] The number of three-rowed n-celled Young tableaux Y such that Y[2, 6] = 15, equals 1001 M(n - 6) - 7007 M(n - 7) + 21021 M(n - 8) - 33033 M(n - 9) + 26026 M(n - 10) - 6006 M(n - 11) - 12012 M(n - 12) + 16016 M(n - 13) + 2002 M(n - 15) The first, 30, non-zero terms are [36036, 93093, 250250, 686686, 1895894, 5267262, 14690676, 41119078, 115437322, 324964640, 917074158, 2593995404, 7352883538, 20883675813, 59423814450, 169382154942, 483595448336, 1382808756329, 3959760306502, 11354478261126, 32600513315370, 93715332605895, 269710642809608, 777071639143800, 2241174953285268, 6470235196896471, 18697063470683598, 54077498103498870, 156542312289308946, 453525959087780579, 1314958881463799572] The number of three-rowed n-celled Young tableaux Y such that Y[2, 7] = 15, equals 1001 M(n - 7) - 6006 M(n - 8) + 13585 M(n - 9) - 16874 M(n - 10) + 9438 M(n - 11) + 13728 M(n - 12) - 18447 M(n - 13) + 3575 M(n - 15) The first, 30, non-zero terms are [5577, 12155, 30888, 81510, 221221, 610467, 1705275, 4801797, 13599586, 38676638, 110340230, 315557242, 904231328, 2595357336, 7459877997, 21468965661, 61855978041, 178403039243, 515039312788, 1488228878994, 4303960951433, 12457112391295, 36082822368736, 104593108668906, 303397428056151, 880673422625997, 2558002048584840, 7434632067875298, 21621255553157975, 62915258046373161, 183179134412684982] The number of three-rowed n-celled Young tableaux Y such that Y[3, 1] = 15, equals 13 M(n - 3) - 66 M(n - 4) + 165 M(n - 5) - 430 M(n - 6) + 786 M(n - 7) - 784 M(n - 8) + 799 M(n - 9) - 670 M(n - 10) + 216 M(n - 11) - 130 M(n - 12) + 36 M(n - 13) + M(n - 15) The first, 30, non-zero terms are [3431, 9851, 29034, 84790, 247758, 722356, 2103697, 6119651, 17788020, 51674520, 150059407, 435671999, 1264811809, 3672066931, 10662348788, 30965923534, 89955711027, 261400435023, 759856602323, 2209612213627, 6427882899139, 18706534288923, 54462336959719, 158627793141675, 462214542972065, 1347378037797701, 3929309655198771, 11463644779925699, 33458546920539395, 97693827962434383, 285364770103297681] The number of three-rowed n-celled Young tableaux Y such that Y[3, 2] = 15, equals 560 M(n - 6) - 2478 M(n - 7) + 4116 M(n - 8) - 5096 M(n - 9) + 4772 M(n - 10) - 1842 M(n - 11) + 1176 M(n - 12) - 428 M(n - 13) - 12 M(n - 15) The first, 30, non-zero terms are [17512, 51976, 148790, 427726, 1222932, 3492884, 9964936, 28423248, 81083968, 231421568, 660947728, 1889236944, 5405054592, 15478590960, 44370322050, 127317433650, 365692480760, 1051409382616, 3025850098478, 8716315662246, 25131447498740, 72525272202724, 209477048584938, 605545107319346, 1751891968717504, 5072322653163392, 14697157959332306, 42616258936323130, 123657835771748356, 359055504537893012, 1043241449829112234] The number of three-rowed n-celled Young tableaux Y such that Y[3, 3] = 15, equals 5733 M(n - 9) - 16653 M(n - 10) + 12376 M(n - 11) - 4394 M(n - 12) + 780 M(n - 13) + 156 M(n - 15) The first, 30, non-zero terms are [38194, 102414, 283062, 772798, 2119143, 5820204, 16040063, 44348460, 123030219, 342409600, 955922097, 2676495120, 7514545779, 21152261052, 59683956345, 168786696468, 478339509492, 1358294424318, 3864184193220, 11012318013294, 31434847654821, 89869828348300, 257304823114521, 737696843799708, 2117732772065534, 6086916692841894, 17515741311585894, 50459007561809782, 145513682315258673, 420050529863246568, 1213699058998794589] The number of three-rowed n-celled Young tableaux Y such that Y[3, 4] = 15, equals 16016 M(n - 12) - 17017 M(n - 13) + 1001 M(n - 15) The first, 30, non-zero terms are [31031, 77077, 185185, 463463, 1175174, 3033030, 7927920, 20960940, 55950895, 150595445, 408268861, 1113859747, 3055937885, 8426000583, 23336472159, 64892268441, 181104440517, 507107856255, 1424235816003, 4011109451349, 11325357911868, 32052285521256, 90909649520690, 258367122527514, 735665108715537, 2098379999927211, 5995147804833187, 17154685518055085, 49157765746967878, 141055818737327494, 405271009137514520] The number of three-rowed n-celled Young tableaux Y such that Y[3, 5] = 15, equals 6006 M(n - 15) The first, 30, non-zero terms are [6006, 6006, 12012, 24024, 54054, 126126, 306306, 762762, 1939938, 5015010, 13141128, 34822788, 93159066, 251261010, 682485804, 1865295432, 5125922802, 14154814674, 39257510292, 109304899704, 305417226114, 856140639354, 2406983917338, 6785341052490, 19175523148782, 54314675421006, 154172929766856, 438484683448812, 1249387808923254, 3566019165682974, 10194497140093266] this took, 1031.887, second of CPU time