The Sorting Probabilities of The entries in the first row vs. those not rel\ ated to it in lower rows in a random Standard Young tableau of shape, [n, n, n, n, n, n], and its Limiting behavior as n goes to infinity for i from 2 to, 5 By Shalosh B. Ekhad --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 2], vs. those in the, 2, -th row from j=1 to j=, 1, are as follws -6 + n [- --------] -1 + 6 n and in Maple notation [-(-6+n)/(-1+6*n)] The limits, as n goes to infinity are [-1/6] and in Maple notation [-1/6] and in floating point [-.1666666667] The cut off is at j=, 1 The rational functions describing the sorting probabilities of the cell, [1, 2], vs. those in the, 3, -th row from j=1 to j=, 1, are as follws (4 n + 1) (11 n - 17) [- -----------------------] 3 (-1 + 6 n) (-1 + 3 n) and in Maple notation [-1/3*(4*n+1)*(11*n-17)/(-1+6*n)/(-1+3*n)] The limits, as n goes to infinity are -22 [---] 27 and in Maple notation [-22/27] and in floating point [-.8148148148] The cut off is at j=, 1 The rational functions describing the sorting probabilities of the cell, [1, 2], vs. those in the, 4, -th row from j=1 to j=, 1, are as follws 3 2 211 n - 246 n + 11 n - 36 [- ----------------------------------] 6 (-1 + 2 n) (-1 + 6 n) (-1 + 3 n) and in Maple notation [-1/6*(211*n^3-246*n^2+11*n-36)/(-1+2*n)/(-1+6*n)/(-1+3*n)] The limits, as n goes to infinity are -211 [----] 216 and in Maple notation [-211/216] and in floating point [-.9768518519] The cut off is at j=, 1 The rational functions describing the sorting probabilities of the cell, [1, 2], vs. those in the, 5, -th row from j=1 to j=, 1, are as follws 4 3 2 647 n - 1090 n + 595 n - 200 n - 12 [- ---------------------------------------------] 6 (-2 + 3 n) (-1 + 2 n) (-1 + 6 n) (-1 + 3 n) and in Maple notation [-1/6*(647*n^4-1090*n^3+595*n^2-200*n-12)/(-2+3*n)/(-1+2*n)/(-1+6*n)/(-1+3*n)] The limits, as n goes to infinity are -647 [----] 648 and in Maple notation [-647/648] and in floating point [-.9984567901] The cut off is at j=, 1 The rational functions describing the sorting probabilities of the cell, [1, 2], vs. those in the, 6, -th row from j=1 to j=, 1, are as follws 5 4 3 2 23327 n - 58335 n + 54995 n - 24525 n + 4658 n - 480 [- ---------------------------------------------------------] 36 (-1 + 6 n) (-1 + 3 n) (-1 + 2 n) (-2 + 3 n) (-5 + 6 n) and in Maple notation [-1/36*(23327*n^5-58335*n^4+54995*n^3-24525*n^2+4658*n-480)/(-1+6*n)/(-1+3*n)/( -1+2*n)/(-2+3*n)/(-5+6*n)] The limits, as n goes to infinity are -23327 [------] 23328 and in Maple notation [-23327/23328] and in floating point [-.9999571331] The cut off is at j=, 1 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 3], vs. those in the, 2, -th row from j=1 to j=, 2, are as follws 2 26 n + 57 n - 53 [-----------------------, 3 (-1 + 3 n) (-1 + 6 n) 5 4 3 2 22319 n - 130420 n + 211045 n - 138020 n + 36216 n - 2520 - -------------------------------------------------------------] 36 (-7 + 6 n) (-1 + 6 n) (-1 + 2 n) (-2 + 3 n) (-5 + 6 n) and in Maple notation [1/3*(26*n^2+57*n-53)/(-1+3*n)/(-1+6*n), -1/36*(22319*n^5-130420*n^4+211045*n^3 -138020*n^2+36216*n-2520)/(-7+6*n)/(-1+6*n)/(-1+2*n)/(-2+3*n)/(-5+6*n)] The limits, as n goes to infinity are 13 -22319 [--, ------] 27 46656 and in Maple notation [13/27, -22319/46656] and in floating point [.4814814815, -.4783736283] The cut off is at j=, 2 The rational functions describing the sorting probabilities of the cell, [1, 3], vs. those in the, 3, -th row from j=1 to j=, 2, are as follws (-6 + n) (2 n + 1) (31 n - 29) 7 6 5 [- ----------------------------------, - (1193147 n - 6421645 n + 12945857 n 3 (-1 + 6 n) (-1 + 3 n) (-2 + 3 n) 4 3 2 - 13445095 n + 8048048 n - 2811640 n + 509688 n - 30240)/(108 (-4 + 3 n) (-7 + 6 n) (-5 + 6 n) (-2 + 3 n) (-1 + 2 n) (-1 + 3 n) (-1 + 6 n))] and in Maple notation [-1/3*(-6+n)*(2*n+1)*(31*n-29)/(-1+6*n)/(-1+3*n)/(-2+3*n), -1/108*(1193147*n^7-\ 6421645*n^6+12945857*n^5-13445095*n^4+8048048*n^3-2811640*n^2+509688*n-30240)/( -4+3*n)/(-7+6*n)/(-5+6*n)/(-2+3*n)/(-1+2*n)/(-1+3*n)/(-1+6*n)] The limits, as n goes to infinity are -31 -1193147 [---, --------] 81 1259712 and in Maple notation [-31/81, -1193147/1259712] and in floating point [-.3827160494, -.9471585569] The cut off is at j=, 1 The rational functions describing the sorting probabilities of the cell, [1, 3], vs. those in the, 4, -th row from j=1 to j=, 2, are as follws 5 4 3 2 5039 n - 18250 n + 11185 n - 1205 n + 3471 n - 2430 8 [- --------------------------------------------------------, - (2513974 n 9 (-5 + 6 n) (-2 + 3 n) (-1 + 2 n) (-1 + 3 n) (-1 + 6 n) 7 6 5 4 3 - 16415801 n + 44422279 n - 65735531 n + 57426151 n - 29869904 n 2 + 9053616 n - 1498824 n + 90720)/(108 (-3 + 2 n) (-4 + 3 n) (-7 + 6 n) (-5 + 6 n) (-2 + 3 n) (-1 + 2 n) (-1 + 3 n) (-1 + 6 n))] and in Maple notation [-1/9*(5039*n^5-18250*n^4+11185*n^3-1205*n^2+3471*n-2430)/(-5+6*n)/(-2+3*n)/(-1 +2*n)/(-1+3*n)/(-1+6*n), -1/108*(2513974*n^8-16415801*n^7+44422279*n^6-65735531 *n^5+57426151*n^4-29869904*n^3+9053616*n^2-1498824*n+90720)/(-3+2*n)/(-4+3*n)/( -7+6*n)/(-5+6*n)/(-2+3*n)/(-1+2*n)/(-1+3*n)/(-1+6*n)] The limits, as n goes to infinity are -5039 -1256987 [-----, --------] 5832 1259712 and in Maple notation [-5039/5832, -1256987/1259712] and in floating point [-.8640260631, -.9978368071] The cut off is at j=, 1 The rational functions describing the sorting probabilities of the cell, [1, 3], vs. those in the, 5, -th row from j=1 to j=, 2, are as follws 7 6 5 4 3 2 [- (620477 n - 3218095 n + 6302087 n - 6640465 n + 4379138 n - 1660630 n + 38028 n + 156240)/(54 (-4 + 3 n) (-7 + 6 n) (-5 + 6 n) (-2 + 3 n) 9 8 (-1 + 2 n) (-1 + 3 n) (-1 + 6 n)), - (2519343 n - 20576507 n 7 6 5 4 3 + 71796358 n - 139919462 n + 166782367 n - 125460983 n + 59092252 n 2 - 16588128 n + 2518200 n - 151200)/(36 (-5 + 3 n) (-3 + 2 n) (-4 + 3 n) (-7 + 6 n) (-5 + 6 n) (-2 + 3 n) (-1 + 2 n) (-1 + 3 n) (-1 + 6 n))] and in Maple notation [-1/54*(620477*n^7-3218095*n^6+6302087*n^5-6640465*n^4+4379138*n^3-1660630*n^2+ 38028*n+156240)/(-4+3*n)/(-7+6*n)/(-5+6*n)/(-2+3*n)/(-1+2*n)/(-1+3*n)/(-1+6*n), -1/36*(2519343*n^9-20576507*n^8+71796358*n^7-139919462*n^6+166782367*n^5-\ 125460983*n^4+59092252*n^3-16588128*n^2+2518200*n-151200)/(-5+3*n)/(-3+2*n)/(-4 +3*n)/(-7+6*n)/(-5+6*n)/(-2+3*n)/(-1+2*n)/(-1+3*n)/(-1+6*n)] The limits, as n goes to infinity are -620477 -31103 [-------, ------] 629856 31104 and in Maple notation [-620477/629856, -31103/31104] and in floating point [-.9851092948, -.9999678498] The cut off is at j=, 1 The rational functions describing the sorting probabilities of the cell, [1, 3], vs. those in the, 6, -th row from j=1 to j=, 2, are as follws 9 8 7 6 5 [- 5 (755343 n - 6177745 n + 21528014 n - 41949226 n + 50125967 n 4 3 2 - 37692715 n + 17539916 n - 4855134 n + 861300 n - 136080)/(54 (-5 + 3 n) (-3 + 2 n) (-4 + 3 n) (-7 + 6 n) (-5 + 6 n) (-2 + 3 n) 10 9 (-1 + 2 n) (-1 + 3 n) (-1 + 6 n)), - (90699253 n - 906992915 n 8 7 6 5 + 3942895590 n - 9775383270 n + 15238457109 n - 15526119195 n 4 3 2 + 10405840760 n - 4496767180 n + 1187286888 n - 170868240 n + 9979200)/ (216 (-1 + 6 n) (-1 + 3 n) (-1 + 2 n) (-2 + 3 n) (-5 + 6 n) (-7 + 6 n) (-4 + 3 n) (-3 + 2 n) (-5 + 3 n) (-11 + 6 n))] and in Maple notation [-5/54*(755343*n^9-6177745*n^8+21528014*n^7-41949226*n^6+50125967*n^5-37692715* n^4+17539916*n^3-4855134*n^2+861300*n-136080)/(-5+3*n)/(-3+2*n)/(-4+3*n)/(-7+6* n)/(-5+6*n)/(-2+3*n)/(-1+2*n)/(-1+3*n)/(-1+6*n), -1/216*(90699253*n^10-\ 906992915*n^9+3942895590*n^8-9775383270*n^7+15238457109*n^6-15526119195*n^5+ 10405840760*n^4-4496767180*n^3+1187286888*n^2-170868240*n+9979200)/(-1+6*n)/(-1 +3*n)/(-1+2*n)/(-2+3*n)/(-5+6*n)/(-7+6*n)/(-4+3*n)/(-3+2*n)/(-5+3*n)/(-11+6*n)] The limits, as n goes to infinity are -419635 -90699253 [-------, ---------] 419904 90699264 and in Maple notation [-419635/419904, -90699253/90699264] and in floating point [-.9993593774, -.9999998787] The cut off is at j=, 1 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 4], vs. those in the, 2, -th row from j=1 to j=, 3, are as follws 3 2 29 n + 6 n - 66 n + 41 7 6 5 [--------------------------------, (120359 n + 1198955 n - 6461551 n (-1 + 2 n) (-1 + 3 n) (-1 + 6 n) 4 3 2 + 10812545 n - 8305624 n + 3094100 n - 497664 n + 20160)/(72 (-4 + 3 n) (-7 + 6 n) (-5 + 6 n) (-2 + 3 n) (-1 + 2 n) (-1 + 3 n) (-1 + 6 n)), - ( 10 9 8 7 913170673 n - 12661626003 n + 73568573214 n - 238263754782 n 6 5 4 3 + 478094522745 n - 620851215747 n + 526381247096 n - 285390284748 n 2 + 93279454272 n - 16107318720 n + 1037836800)/(1728 (-1 + 6 n) (-1 + 2 n) (-2 + 3 n) (-5 + 6 n) (-7 + 6 n) (-4 + 3 n) (-3 + 2 n) (-5 + 3 n) (-11 + 6 n) (-13 + 6 n))] and in Maple notation [(29*n^3+6*n^2-66*n+41)/(-1+2*n)/(-1+3*n)/(-1+6*n), 1/72*(120359*n^7+1198955*n^ 6-6461551*n^5+10812545*n^4-8305624*n^3+3094100*n^2-497664*n+20160)/(-4+3*n)/(-7 +6*n)/(-5+6*n)/(-2+3*n)/(-1+2*n)/(-1+3*n)/(-1+6*n), -1/1728*(913170673*n^10-\ 12661626003*n^9+73568573214*n^8-238263754782*n^7+478094522745*n^6-620851215747* n^5+526381247096*n^4-285390284748*n^3+93279454272*n^2-16107318720*n+1037836800) /(-1+6*n)/(-1+2*n)/(-2+3*n)/(-5+6*n)/(-7+6*n)/(-4+3*n)/(-3+2*n)/(-5+3*n)/(-11+6 *n)/(-13+6*n)] The limits, as n goes to infinity are 29 120359 -913170673 [--, ------, ----------] 36 839808 1451188224 and in Maple notation [29/36, 120359/839808, -913170673/1451188224] and in floating point [.8055555556, .1433172820, -.6292572238] The cut off is at j=, 3 The rational functions describing the sorting probabilities of the cell, [1, 4], vs. those in the, 3, -th row from j=1 to j=, 3, are as follws 5 4 3 2 5 (43 n + 1072 n - 2617 n + 827 n + 1191 n - 510) 9 [--------------------------------------------------------, - (1238186 n 3 (-5 + 6 n) (-2 + 3 n) (-1 + 2 n) (-1 + 3 n) (-1 + 6 n) 8 7 6 5 4 - 12984489 n + 50267426 n - 97662144 n + 108705864 n - 77118971 n 3 2 + 38862764 n - 13949116 n + 2713440 n - 100800)/(24 (-5 + 3 n) (-3 + 2 n) (-4 + 3 n) (-7 + 6 n) (-5 + 6 n) (-2 + 3 n) (-1 + 2 n) 12 11 (-1 + 3 n) (-1 + 6 n)), - (6391923361 n - 94093822437 n 10 9 8 + 610692259177 n - 2314482215565 n + 5703675628743 n 7 6 5 - 9611418228531 n + 11319621462851 n - 9344052464895 n 4 3 2 + 5332681123196 n - 2036026757532 n + 488794012272 n - 65417768640 n + 3632428800)/(864 (-7 + 3 n) (-13 + 6 n) (-11 + 6 n) (-5 + 3 n) (-3 + 2 n) (-4 + 3 n) (-7 + 6 n) (-5 + 6 n) (-2 + 3 n) (-1 + 2 n) (-1 + 3 n) (-1 + 6 n))] and in Maple notation [5/3*(43*n^5+1072*n^4-2617*n^3+827*n^2+1191*n-510)/(-5+6*n)/(-2+3*n)/(-1+2*n)/( -1+3*n)/(-1+6*n), -1/24*(1238186*n^9-12984489*n^8+50267426*n^7-97662144*n^6+ 108705864*n^5-77118971*n^4+38862764*n^3-13949116*n^2+2713440*n-100800)/(-5+3*n) /(-3+2*n)/(-4+3*n)/(-7+6*n)/(-5+6*n)/(-2+3*n)/(-1+2*n)/(-1+3*n)/(-1+6*n), -1/ 864*(6391923361*n^12-94093822437*n^11+610692259177*n^10-2314482215565*n^9+ 5703675628743*n^8-9611418228531*n^7+11319621462851*n^6-9344052464895*n^5+ 5332681123196*n^4-2036026757532*n^3+488794012272*n^2-65417768640*n+3632428800)/ (-7+3*n)/(-13+6*n)/(-11+6*n)/(-5+3*n)/(-3+2*n)/(-4+3*n)/(-7+6*n)/(-5+6*n)/(-2+3 *n)/(-1+2*n)/(-1+3*n)/(-1+6*n)] The limits, as n goes to infinity are 215 -619093 -6391923361 [----, -------, -----------] 1944 839808 6530347008 and in Maple notation [215/1944, -619093/839808, -6391923361/6530347008] and in floating point [.1105967078, -.7371839754, -.9788030182] The cut off is at j=, 2 The rational functions describing the sorting probabilities of the cell, [1, 4], vs. those in the, 4, -th row from j=1 to j=, 3, are as follws [- 5 (-6 + n) 6 5 4 3 2 (3869 n - 12835 n + 9903 n + 1599 n + 1356 n - 5632 n + 2324)/(4 (-3 + 2 n) (-7 + 6 n) (-5 + 6 n) (-2 + 3 n) (-1 + 2 n) (-1 + 3 n) 10 9 8 7 (-1 + 6 n)), - (88622741 n - 915076125 n + 3954091605 n - 9715293120 n 6 5 4 3 + 15220528173 n - 15683275695 n + 10412700445 n - 4334232180 n 2 + 1188988236 n - 228502080 n + 9979200)/(216 (-1 + 6 n) (-1 + 3 n) (-1 + 2 n) (-2 + 3 n) (-5 + 6 n) (-7 + 6 n) (-4 + 3 n) (-3 + 2 n) 13 12 (-5 + 3 n) (-11 + 6 n)), - (78334059241 n - 1332295473717 n 11 10 9 + 10170190067567 n - 46059684653685 n + 137738632914213 n 8 7 6 - 286352439577731 n + 424320064011581 n - 451949851574055 n 5 4 3 + 344365042413646 n - 184314543032652 n + 66872218117752 n 2 - 15432773067360 n + 2006122204800 n - 108972864000)/(5184 (-5 + 2 n) (-7 + 3 n) (-13 + 6 n) (-1 + 6 n) (-1 + 3 n) (-1 + 2 n) (-2 + 3 n) (-5 + 6 n) (-7 + 6 n) (-4 + 3 n) (-3 + 2 n) (-5 + 3 n) (-11 + 6 n))] and in Maple notation [-5/4*(-6+n)*(3869*n^6-12835*n^5+9903*n^4+1599*n^3+1356*n^2-5632*n+2324)/(-3+2* n)/(-7+6*n)/(-5+6*n)/(-2+3*n)/(-1+2*n)/(-1+3*n)/(-1+6*n), -1/216*(88622741*n^10 -915076125*n^9+3954091605*n^8-9715293120*n^7+15220528173*n^6-15683275695*n^5+ 10412700445*n^4-4334232180*n^3+1188988236*n^2-228502080*n+9979200)/(-1+6*n)/(-1 +3*n)/(-1+2*n)/(-2+3*n)/(-5+6*n)/(-7+6*n)/(-4+3*n)/(-3+2*n)/(-5+3*n)/(-11+6*n), -1/5184*(78334059241*n^13-1332295473717*n^12+10170190067567*n^11-46059684653685 *n^10+137738632914213*n^9-286352439577731*n^8+424320064011581*n^7-\ 451949851574055*n^6+344365042413646*n^5-184314543032652*n^4+66872218117752*n^3-\ 15432773067360*n^2+2006122204800*n-108972864000)/(-5+2*n)/(-7+3*n)/(-13+6*n)/(-\ 1+6*n)/(-1+3*n)/(-1+2*n)/(-2+3*n)/(-5+6*n)/(-7+6*n)/(-4+3*n)/(-3+2*n)/(-5+3*n)/ (-11+6*n)] The limits, as n goes to infinity are -19345 -88622741 -78334059241 [------, ---------, ------------] 31104 90699264 78364164096 and in Maple notation [-19345/31104, -88622741/90699264, -78334059241/78364164096] and in floating point [-.6219457305, -.9771054041, -.9996158339] The cut off is at j=, 1 The rational functions describing the sorting probabilities of the cell, [1, 4], vs. those in the, 5, -th row from j=1 to j=, 3, are as follws 10 9 8 7 6 [- (28339163 n - 307234365 n + 1345854030 n - 3219607410 n + 4862992539 n 5 4 3 2 - 5168953245 n + 4015304020 n - 1934593140 n + 163962648 n + 340000560 n - 123076800)/(72 (-1 + 6 n) (-1 + 3 n) (-1 + 2 n) (-2 + 3 n) (-5 + 6 n) (-7 + 6 n) (-4 + 3 n) (-3 + 2 n) (-5 + 3 n) (-11 + 6 n)), - ( 12 11 10 9 3262964891 n - 47366073372 n + 305344274627 n - 1155567638130 n 8 7 6 + 2850389315583 n - 4807020681456 n + 5660644241941 n 5 4 3 2 - 4674019869930 n + 2669801961526 n - 1014175414872 n + 241292515032 n - 34788882240 n + 1816214400)/(432 (-1 + 6 n) (-1 + 3 n) (-1 + 2 n) (-2 + 3 n) (-5 + 6 n) (-7 + 6 n) (-4 + 3 n) (-3 + 2 n) (-5 + 3 n) 14 13 (-11 + 6 n) (-13 + 6 n) (-7 + 3 n)), - (78364014089 n - 1541164122257 n 12 11 10 + 13722423449331 n - 73180546707793 n + 260567995835017 n 9 8 7 - 653653732789671 n + 1187916308266393 n - 1583473232119579 n 6 5 4 + 1549579319121594 n - 1102618874410172 n + 558367451767976 n 3 2 - 193759321453728 n + 43162974345600 n - 5458632076800 n + 290594304000) /(1728 (-8 + 3 n) (-5 + 2 n) (-1 + 6 n) (-1 + 3 n) (-1 + 2 n) (-2 + 3 n) (-5 + 6 n) (-7 + 6 n) (-4 + 3 n) (-3 + 2 n) (-5 + 3 n) (-11 + 6 n) (-13 + 6 n) (-7 + 3 n))] and in Maple notation [-1/72*(28339163*n^10-307234365*n^9+1345854030*n^8-3219607410*n^7+4862992539*n^ 6-5168953245*n^5+4015304020*n^4-1934593140*n^3+163962648*n^2+340000560*n-\ 123076800)/(-1+6*n)/(-1+3*n)/(-1+2*n)/(-2+3*n)/(-5+6*n)/(-7+6*n)/(-4+3*n)/(-3+2 *n)/(-5+3*n)/(-11+6*n), -1/432*(3262964891*n^12-47366073372*n^11+305344274627*n ^10-1155567638130*n^9+2850389315583*n^8-4807020681456*n^7+5660644241941*n^6-\ 4674019869930*n^5+2669801961526*n^4-1014175414872*n^3+241292515032*n^2-\ 34788882240*n+1816214400)/(-1+6*n)/(-1+3*n)/(-1+2*n)/(-2+3*n)/(-5+6*n)/(-7+6*n) /(-4+3*n)/(-3+2*n)/(-5+3*n)/(-11+6*n)/(-13+6*n)/(-7+3*n), -1/1728*(78364014089* n^14-1541164122257*n^13+13722423449331*n^12-73180546707793*n^11+260567995835017 *n^10-653653732789671*n^9+1187916308266393*n^8-1583473232119579*n^7+ 1549579319121594*n^6-1102618874410172*n^5+558367451767976*n^4-193759321453728*n ^3+43162974345600*n^2-5458632076800*n+290594304000)/(-8+3*n)/(-5+2*n)/(-1+6*n)/ (-1+3*n)/(-1+2*n)/(-2+3*n)/(-5+6*n)/(-7+6*n)/(-4+3*n)/(-3+2*n)/(-5+3*n)/(-11+6* n)/(-13+6*n)/(-7+3*n)] The limits, as n goes to infinity are -28339163 -3262964891 -78364014089 [---------, -----------, ------------] 30233088 3265173504 78364164096 and in Maple notation [-28339163/30233088, -3262964891/3265173504, -78364014089/78364164096] and in floating point [-.9373558864, -.9993235848, -.9999980858] The cut off is at j=, 1 The rational functions describing the sorting probabilities of the cell, [1, 4], vs. those in the, 6, -th row from j=1 to j=, 3, are as follws 13 12 11 10 [- (26018730403 n - 444349694631 n + 3393448634621 n - 15349589299575 n 9 8 7 + 45863887208199 n - 95434503072633 n + 141734609826263 n 6 5 4 - 151052473687245 n + 114304493881498 n - 60055198774236 n 3 2 + 21660061683816 n - 5890180870080 n + 1534311676800 n - 290594304000)/( 1728 (-5 + 2 n) (-7 + 3 n) (-13 + 6 n) (-11 + 6 n) (-5 + 3 n) (-3 + 2 n) (-4 + 3 n) (-7 + 6 n) (-5 + 6 n) (-2 + 3 n) (-1 + 2 n) (-1 + 3 n) 14 13 12 (-1 + 6 n)), - 5 (47018273919 n - 924701016695 n + 8233439141085 n 11 10 9 - 43908334466311 n + 156341034070143 n - 392191570956153 n 8 7 6 + 712749100799559 n - 950088885230365 n + 929744799124998 n 5 4 3 - 661558557160172 n + 335036958482136 n - 116263866564864 n 2 + 25875211656960 n - 3284727344640 n + 174356582400)/(5184 (-8 + 3 n) (-5 + 2 n) (-7 + 3 n) (-13 + 6 n) (-11 + 6 n) (-5 + 3 n) (-3 + 2 n) (-4 + 3 n) (-7 + 6 n) (-5 + 6 n) (-2 + 3 n) (-1 + 2 n) (-1 + 3 n) 15 14 (-1 + 6 n)), - (1410554951297 n - 31737486531810 n 13 12 11 + 325603100846480 n - 2017093591415760 n + 8422430120788094 n 10 9 8 - 25054737607542060 n + 54718837859447740 n - 89086235165834280 n 7 6 5 + 108649569247045321 n - 98875704639850050 n + 66284154265943980 n 4 3 2 - 31964405181771960 n + 10658672365140288 n - 2299562070094080 n + 283620933388800 n - 14820309504000)/(5184 (-1 + 6 n) (-1 + 3 n) (-1 + 2 n) (-2 + 3 n) (-5 + 6 n) (-7 + 6 n) (-4 + 3 n) (-3 + 2 n) (-5 + 3 n) (-11 + 6 n) (-13 + 6 n) (-7 + 3 n) (-5 + 2 n) (-8 + 3 n) (-17 + 6 n))] and in Maple notation [-1/1728*(26018730403*n^13-444349694631*n^12+3393448634621*n^11-15349589299575* n^10+45863887208199*n^9-95434503072633*n^8+141734609826263*n^7-151052473687245* n^6+114304493881498*n^5-60055198774236*n^4+21660061683816*n^3-5890180870080*n^2 +1534311676800*n-290594304000)/(-5+2*n)/(-7+3*n)/(-13+6*n)/(-11+6*n)/(-5+3*n)/( -3+2*n)/(-4+3*n)/(-7+6*n)/(-5+6*n)/(-2+3*n)/(-1+2*n)/(-1+3*n)/(-1+6*n), -5/5184 *(47018273919*n^14-924701016695*n^13+8233439141085*n^12-43908334466311*n^11+ 156341034070143*n^10-392191570956153*n^9+712749100799559*n^8-950088885230365*n^ 7+929744799124998*n^6-661558557160172*n^5+335036958482136*n^4-116263866564864*n ^3+25875211656960*n^2-3284727344640*n+174356582400)/(-8+3*n)/(-5+2*n)/(-7+3*n)/ (-13+6*n)/(-11+6*n)/(-5+3*n)/(-3+2*n)/(-4+3*n)/(-7+6*n)/(-5+6*n)/(-2+3*n)/(-1+2 *n)/(-1+3*n)/(-1+6*n), -1/5184*(1410554951297*n^15-31737486531810*n^14+ 325603100846480*n^13-2017093591415760*n^12+8422430120788094*n^11-\ 25054737607542060*n^10+54718837859447740*n^9-89086235165834280*n^8+ 108649569247045321*n^7-98875704639850050*n^6+66284154265943980*n^5-\ 31964405181771960*n^4+10658672365140288*n^3-2299562070094080*n^2+ 283620933388800*n-14820309504000)/(-1+6*n)/(-1+3*n)/(-1+2*n)/(-2+3*n)/(-5+6*n)/ (-7+6*n)/(-4+3*n)/(-3+2*n)/(-5+3*n)/(-11+6*n)/(-13+6*n)/(-7+3*n)/(-5+2*n)/(-8+3 *n)/(-17+6*n)] The limits, as n goes to infinity are -26018730403 -78363789865 -1410554951297 [------------, ------------, --------------] 26121388032 78364164096 1410554953728 and in Maple notation [-26018730403/26121388032, -78363789865/78364164096, -1410554951297/ 1410554953728] and in floating point [-.9960699780, -.9999952245, -.9999999983] The cut off is at j=, 1 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 5], vs. those in the, 2, -th row from j=1 to j=, 4, are as follws 4 3 2 101 n - 110 n - 140 n + 325 n - 166 8 7 [-------------------------------------------, 35 (42034 n - 121811 n (-2 + 3 n) (-1 + 2 n) (-1 + 3 n) (-1 + 6 n) 6 5 4 3 2 - 268631 n + 1517779 n - 2379695 n + 1751908 n - 633312 n + 96408 n - 2592)/(108 (-1 + 6 n) (-1 + 3 n) (-1 + 2 n) (-2 + 3 n) (-5 + 6 n) 12 11 (-7 + 6 n) (-4 + 3 n) (-3 + 2 n)), - (5036265376 n - 231026679927 n 10 9 8 + 2481674440952 n - 12931902063435 n + 39892084857708 n 7 6 5 - 79301974787721 n + 105580287404576 n - 95271444098865 n 4 3 2 + 57741542731516 n - 22760521442052 n + 5456081986272 n - 692552246400 n + 32691859200)/(7776 (-7 + 3 n) (-13 + 6 n) (-11 + 6 n) (-5 + 3 n) (-1 + 6 n) (-1 + 3 n) (-1 + 2 n) (-2 + 3 n) (-5 + 6 n) 15 (-7 + 6 n) (-4 + 3 n) (-3 + 2 n)), - (18229731632639 n 14 13 12 - 494851062520413 n + 6077743615210094 n - 44838029630779887 n 11 10 9 + 222224739308617964 n - 783342026582186769 n + 2026636730575099102 n 8 7 6 - 3912142681255912821 n + 5667924428932395565 n - 6143963091365586678 n 5 4 3 + 4921497726209746204 n - 2844269100432545592 n + 1138391926214159232 n 2 - 294032737114467840 n + 42845125674355200 n - 2534272925184000)/(46656 (-5 + 2 n) (-1 + 2 n) (-17 + 6 n) (-5 + 6 n) (-4 + 3 n) (-19 + 6 n) (-11 + 6 n) (-7 + 3 n) (-2 + 3 n) (-5 + 3 n) (-8 + 3 n) (-7 + 6 n) (-13 + 6 n) (-1 + 6 n) (-3 + 2 n))] and in Maple notation [(101*n^4-110*n^3-140*n^2+325*n-166)/(-2+3*n)/(-1+2*n)/(-1+3*n)/(-1+6*n), 35/ 108*(42034*n^8-121811*n^7-268631*n^6+1517779*n^5-2379695*n^4+1751908*n^3-633312 *n^2+96408*n-2592)/(-1+6*n)/(-1+3*n)/(-1+2*n)/(-2+3*n)/(-5+6*n)/(-7+6*n)/(-4+3* n)/(-3+2*n), -1/7776*(5036265376*n^12-231026679927*n^11+2481674440952*n^10-\ 12931902063435*n^9+39892084857708*n^8-79301974787721*n^7+105580287404576*n^6-\ 95271444098865*n^5+57741542731516*n^4-22760521442052*n^3+5456081986272*n^2-\ 692552246400*n+32691859200)/(-7+3*n)/(-13+6*n)/(-11+6*n)/(-5+3*n)/(-1+6*n)/(-1+ 3*n)/(-1+2*n)/(-2+3*n)/(-5+6*n)/(-7+6*n)/(-4+3*n)/(-3+2*n), -1/46656*( 18229731632639*n^15-494851062520413*n^14+6077743615210094*n^13-\ 44838029630779887*n^12+222224739308617964*n^11-783342026582186769*n^10+ 2026636730575099102*n^9-3912142681255912821*n^8+5667924428932395565*n^7-\ 6143963091365586678*n^6+4921497726209746204*n^5-2844269100432545592*n^4+ 1138391926214159232*n^3-294032737114467840*n^2+42845125674355200*n-\ 2534272925184000)/(-5+2*n)/(-1+2*n)/(-17+6*n)/(-5+6*n)/(-4+3*n)/(-19+6*n)/(-11+ 6*n)/(-7+3*n)/(-2+3*n)/(-5+3*n)/(-8+3*n)/(-7+6*n)/(-13+6*n)/(-1+6*n)/(-3+2*n)] The limits, as n goes to infinity are 101 735595 -157383293 -18229731632639 [---, -------, ----------, ---------------] 108 1259712 1836660096 25389989167104 and in Maple notation [101/108, 735595/1259712, -157383293/1836660096, -18229731632639/25389989167104 ] and in floating point [.9351851852, .5839390273, -.8568993977e-1, -.7179889488] The cut off is at j=, 3 The rational functions describing the sorting probabilities of the cell, [1, 5], vs. those in the, 3, -th row from j=1 to j=, 4, are as follws 7 6 5 4 3 2 [(105659 n + 120815 n - 2729431 n + 6735185 n - 4880734 n - 2067910 n + 3732156 n - 992880)/(18 (-1 + 6 n) (-1 + 3 n) (-1 + 2 n) (-2 + 3 n) 10 9 (-5 + 6 n) (-7 + 6 n) (-4 + 3 n)), - 7 (9565423 n - 189862305 n 8 7 6 5 + 1143143470 n - 3182365490 n + 4694512639 n - 3971413745 n 4 3 2 + 2310518580 n - 1319427260 n + 661590288 n - 157975200 n + 2851200)/( 432 (-1 + 6 n) (-1 + 3 n) (-1 + 2 n) (-2 + 3 n) (-5 + 6 n) (-7 + 6 n) 14 (-4 + 3 n) (-3 + 2 n) (-5 + 3 n) (-11 + 6 n)), - (102094804561 n 13 12 11 - 2159151815283 n + 20081901558954 n - 109525004557227 n 10 9 8 + 393089097568628 n - 985472348769009 n + 1783610204357042 n 7 6 5 - 2368987421161521 n + 2316745579118211 n - 1653143387239008 n 4 3 2 + 841244474535004 n - 293052993862752 n + 65218903257600 n - 8187948115200 n + 435891456000)/(2592 (-3 + 2 n) (-1 + 6 n) (-1 + 3 n) (-13 + 6 n) (-7 + 6 n) (-8 + 3 n) (-5 + 3 n) (-2 + 3 n) (-7 + 3 n) (-11 + 6 n) (-4 + 3 n) (-5 + 6 n) (-1 + 2 n) (-5 + 2 n)), - ( 17 16 15 904713369297851 n - 26363124116005104 n + 353412654614804880 n 14 13 - 2892842975117938620 n + 16178049703378865022 n 12 11 - 65515326790642571388 n + 198627784139434720040 n 10 9 - 459615354867591873060 n + 819911941406401753503 n 8 7 - 1130996335165143749412 n + 1202517266484522082080 n 6 5 - 976198921594618863120 n + 594951980395617068624 n 4 3 - 265047175967113908096 n + 82784551923235872000 n 2 - 16934582501186227200 n + 2002390328010240000 n - 101370917007360000)/( 186624 (-3 + 2 n) (-1 + 6 n) (-1 + 3 n) (-13 + 6 n) (-7 + 6 n) (-10 + 3 n) (-8 + 3 n) (-5 + 3 n) (-2 + 3 n) (-7 + 3 n) (-11 + 6 n) (-19 + 6 n) (-4 + 3 n) (-5 + 6 n) (-17 + 6 n) (-1 + 2 n) (-5 + 2 n))] and in Maple notation [1/18*(105659*n^7+120815*n^6-2729431*n^5+6735185*n^4-4880734*n^3-2067910*n^2+ 3732156*n-992880)/(-1+6*n)/(-1+3*n)/(-1+2*n)/(-2+3*n)/(-5+6*n)/(-7+6*n)/(-4+3*n ), -7/432*(9565423*n^10-189862305*n^9+1143143470*n^8-3182365490*n^7+4694512639* n^6-3971413745*n^5+2310518580*n^4-1319427260*n^3+661590288*n^2-157975200*n+ 2851200)/(-1+6*n)/(-1+3*n)/(-1+2*n)/(-2+3*n)/(-5+6*n)/(-7+6*n)/(-4+3*n)/(-3+2*n )/(-5+3*n)/(-11+6*n), -1/2592*(102094804561*n^14-2159151815283*n^13+ 20081901558954*n^12-109525004557227*n^11+393089097568628*n^10-985472348769009*n ^9+1783610204357042*n^8-2368987421161521*n^7+2316745579118211*n^6-\ 1653143387239008*n^5+841244474535004*n^4-293052993862752*n^3+65218903257600*n^2 -8187948115200*n+435891456000)/(-3+2*n)/(-1+6*n)/(-1+3*n)/(-13+6*n)/(-7+6*n)/(-\ 8+3*n)/(-5+3*n)/(-2+3*n)/(-7+3*n)/(-11+6*n)/(-4+3*n)/(-5+6*n)/(-1+2*n)/(-5+2*n) , -1/186624*(904713369297851*n^17-26363124116005104*n^16+353412654614804880*n^ 15-2892842975117938620*n^14+16178049703378865022*n^13-65515326790642571388*n^12 +198627784139434720040*n^11-459615354867591873060*n^10+819911941406401753503*n^ 9-1130996335165143749412*n^8+1202517266484522082080*n^7-976198921594618863120*n ^6+594951980395617068624*n^5-265047175967113908096*n^4+82784551923235872000*n^3 -16934582501186227200*n^2+2002390328010240000*n-101370917007360000)/(-3+2*n)/(-\ 1+6*n)/(-1+3*n)/(-13+6*n)/(-7+6*n)/(-10+3*n)/(-8+3*n)/(-5+3*n)/(-2+3*n)/(-7+3*n )/(-11+6*n)/(-19+6*n)/(-4+3*n)/(-5+6*n)/(-17+6*n)/(-1+2*n)/(-5+2*n)] The limits, as n goes to infinity are 105659 -66957961 -102094804561 -904713369297851 [------, ---------, -------------, ----------------] 209952 181398528 117546246144 914039610015744 and in Maple notation [105659/209952, -66957961/181398528, -102094804561/117546246144, -\ 904713369297851/914039610015744] and in floating point [.5032531245, -.3691207516, -.8685501061, -.9897966777] The cut off is at j=, 2 The rational functions describing the sorting probabilities of the cell, [1, 5], vs. those in the, 4, -th row from j=1 to j=, 4, are as follws 10 9 8 7 6 [- 7 (613739 n - 15685155 n + 106012050 n - 311174010 n + 414795927 n 5 4 3 2 - 152656575 n - 123768200 n - 47743140 n + 289933884 n - 207189720 n + 48232800)/(36 (-1 + 6 n) (-1 + 3 n) (-1 + 2 n) (-2 + 3 n) (-5 + 6 n) (-7 + 6 n) (-4 + 3 n) (-3 + 2 n) (-5 + 3 n) (-11 + 6 n)), - 7 ( 13 12 11 10 5040882143 n - 93206240531 n + 735352647061 n - 3327135321655 n 9 8 7 + 9801184505619 n - 20213589821493 n + 30294725097703 n 6 5 4 - 32938740425605 n + 24941126969738 n - 12434007547676 n 3 2 + 4238828763336 n - 1381444547040 n + 380773742400 n - 7783776000)/(2592 (-1 + 6 n) (-1 + 3 n) (-1 + 2 n) (-2 + 3 n) (-5 + 6 n) (-7 + 6 n) (-4 + 3 n) (-3 + 2 n) (-5 + 3 n) (-11 + 6 n) (-13 + 6 n) (-7 + 3 n) 15 14 (-5 + 2 n)), - (6311671744733 n - 142751251168005 n 13 12 11 + 1465928555073680 n - 9078288333761790 n + 37895815907310566 n 10 9 8 - 112736606640001080 n + 246251667216665440 n - 400919273185905870 n 7 6 5 + 488896126231799869 n - 444893020007992875 n + 298298945872419280 n 4 3 2 - 143873948413355340 n + 47958253358720832 n - 10338753011495040 n + 1276294200249600 n - 66691392768000)/(23328 (-17 + 6 n) (-8 + 3 n) (-1 + 6 n) (-1 + 3 n) (-1 + 2 n) (-2 + 3 n) (-5 + 6 n) (-7 + 6 n) (-4 + 3 n) (-3 + 2 n) (-5 + 3 n) (-11 + 6 n) (-13 + 6 n) (-7 + 3 n) 18 17 (-5 + 2 n)), - (1827902837843792 n - 59411641887711165 n 16 15 + 894186674232413448 n - 8271441684200361600 n 14 13 + 52630930253085193044 n - 244281220013816795250 n 12 11 + 855749097541812672836 n - 2309340877737743425080 n 10 9 + 4856872764880311862236 n - 8001497196293310685305 n 8 7 + 10322531365636006213284 n - 10370480802130417656240 n 6 5 + 8023331042565503302928 n - 4694513740888939895280 n 4 3 + 2020696349834699586432 n - 613290355524766990080 n 2 + 122537319107398272000 n - 14219474130086400000 n + 709596419051520000)/ (186624 (-3 + 2 n) (-1 + 6 n) (-1 + 3 n) (-13 + 6 n) (-7 + 6 n) (-10 + 3 n) (-8 + 3 n) (-5 + 3 n) (-2 + 3 n) (-7 + 3 n) (-11 + 6 n) (-19 + 6 n) (-4 + 3 n) (-5 + 6 n) (-17 + 6 n) (-7 + 2 n) (-1 + 2 n) (-5 + 2 n))] and in Maple notation [-7/36*(613739*n^10-15685155*n^9+106012050*n^8-311174010*n^7+414795927*n^6-\ 152656575*n^5-123768200*n^4-47743140*n^3+289933884*n^2-207189720*n+48232800)/(-\ 1+6*n)/(-1+3*n)/(-1+2*n)/(-2+3*n)/(-5+6*n)/(-7+6*n)/(-4+3*n)/(-3+2*n)/(-5+3*n)/ (-11+6*n), -7/2592*(5040882143*n^13-93206240531*n^12+735352647061*n^11-\ 3327135321655*n^10+9801184505619*n^9-20213589821493*n^8+30294725097703*n^7-\ 32938740425605*n^6+24941126969738*n^5-12434007547676*n^4+4238828763336*n^3-\ 1381444547040*n^2+380773742400*n-7783776000)/(-1+6*n)/(-1+3*n)/(-1+2*n)/(-2+3*n )/(-5+6*n)/(-7+6*n)/(-4+3*n)/(-3+2*n)/(-5+3*n)/(-11+6*n)/(-13+6*n)/(-7+3*n)/(-5 +2*n), -1/23328*(6311671744733*n^15-142751251168005*n^14+1465928555073680*n^13-\ 9078288333761790*n^12+37895815907310566*n^11-112736606640001080*n^10+ 246251667216665440*n^9-400919273185905870*n^8+488896126231799869*n^7-\ 444893020007992875*n^6+298298945872419280*n^5-143873948413355340*n^4+ 47958253358720832*n^3-10338753011495040*n^2+1276294200249600*n-66691392768000)/ (-17+6*n)/(-8+3*n)/(-1+6*n)/(-1+3*n)/(-1+2*n)/(-2+3*n)/(-5+6*n)/(-7+6*n)/(-4+3* n)/(-3+2*n)/(-5+3*n)/(-11+6*n)/(-13+6*n)/(-7+3*n)/(-5+2*n), -1/186624*( 1827902837843792*n^18-59411641887711165*n^17+894186674232413448*n^16-\ 8271441684200361600*n^15+52630930253085193044*n^14-244281220013816795250*n^13+ 855749097541812672836*n^12-2309340877737743425080*n^11+4856872764880311862236*n ^10-8001497196293310685305*n^9+10322531365636006213284*n^8-\ 10370480802130417656240*n^7+8023331042565503302928*n^6-4694513740888939895280*n ^5+2020696349834699586432*n^4-613290355524766990080*n^3+122537319107398272000*n ^2-14219474130086400000*n+709596419051520000)/(-3+2*n)/(-1+6*n)/(-1+3*n)/(-13+6 *n)/(-7+6*n)/(-10+3*n)/(-8+3*n)/(-5+3*n)/(-2+3*n)/(-7+3*n)/(-11+6*n)/(-19+6*n)/ (-4+3*n)/(-5+6*n)/(-17+6*n)/(-7+2*n)/(-1+2*n)/(-5+2*n)] The limits, as n goes to infinity are -4296173 -35286175001 -6311671744733 -114243927365237 [--------, ------------, --------------, ----------------] 15116544 39182082048 6347497291776 114254951251968 and in Maple notation [-4296173/15116544, -35286175001/39182082048, -6311671744733/6347497291776, -\ 114243927365237/114254951251968] and in floating point [-.2842033867, -.9005691672, -.9943559571, -.9999035150] The cut off is at j=, 1 The rational functions describing the sorting probabilities of the cell, [1, 5], vs. those in the, 5, -th row from j=1 to j=, 4, are as follws 12 11 10 [- (-6 + n) (8140173088 n - 111492692151 n + 656024285066 n 9 8 7 - 2194841122395 n + 4726631213754 n - 7237565748693 n 6 5 4 + 8700320858798 n - 8434277739705 n + 5661505068958 n 3 2 - 1444641754656 n - 1079176655664 n + 987025435200 n - 234897062400)/( 432 (-1 + 2 n) (-5 + 6 n) (-4 + 3 n) (-11 + 6 n) (-7 + 3 n) (-2 + 3 n) (-5 + 3 n) (-8 + 3 n) (-7 + 6 n) (-13 + 6 n) (-1 + 3 n) (-1 + 6 n) 15 14 (-3 + 2 n)), - (2104766143624 n - 47627214026475 n 13 12 11 + 488795392979485 n - 3025124682496785 n + 12627976835537773 n 10 9 8 - 37586670240069555 n + 82120994094584555 n - 133612606028709555 n 7 6 5 + 162798073490408807 n - 148331514708096390 n + 99814924682362460 n 4 3 2 - 47968841564051160 n + 15573122735167296 n - 3417337339606080 n + 594947511648000 n - 22230464256000)/(7776 (-3 + 2 n) (-1 + 6 n) (-1 + 3 n) (-13 + 6 n) (-7 + 6 n) (-8 + 3 n) (-5 + 3 n) (-2 + 3 n) (-7 + 3 n) (-11 + 6 n) (-4 + 3 n) (-5 + 6 n) (-17 + 6 n) (-1 + 2 n) 17 16 (-5 + 2 n)), - (228495560674148 n - 6626881465806852 n 15 14 13 + 88579572156814815 n - 723897701111039985 n + 4045214584187791791 n 12 11 - 16376944780518197589 n + 49649418272449484465 n 10 9 - 114894364857385503555 n + 204978474965297469249 n 8 7 - 282763634392819583091 n + 300644189272431839100 n 6 5 - 244052759862715149360 n + 148732057879868609312 n 4 3 - 66254987497487989968 n + 20693309813190933120 n 2 - 4233509772879513600 n + 500597582002560000 n - 25342729251840000)/( 46656 (-3 + 2 n) (-1 + 6 n) (-1 + 3 n) (-13 + 6 n) (-7 + 6 n) (-10 + 3 n) (-8 + 3 n) (-5 + 3 n) (-2 + 3 n) (-7 + 3 n) (-11 + 6 n) (-19 + 6 n) (-4 + 3 n) (-5 + 6 n) (-17 + 6 n) (-1 + 2 n) (-5 + 2 n)), - ( 19 18 17 5484236552343677 n - 198346606185984214 n + 3336092223077677293 n 16 15 - 34650302010934018956 n + 248878437911469111114 n 14 13 - 1311784196618066653188 n + 5254342404401313571106 n 12 11 - 16341263153211233064352 n + 39973362649052989919961 n 10 9 - 77430086862436662302982 n + 118984069950522233477409 n 8 7 - 144659298175675701727428 n + 138145279824914851070528 n 6 5 - 102340173735079242333616 n + 57701738348286914046192 n 4 3 - 24067533584280604443264 n + 7113806966704179582720 n 2 - 1390568932571640192000 n + 158543004688104960000 n - 7805560609566720000)/(186624 (-1 + 6 n) (-1 + 3 n) (-1 + 2 n) (-2 + 3 n) (-5 + 6 n) (-7 + 6 n) (-4 + 3 n) (-3 + 2 n) (-5 + 3 n) (-11 + 6 n) (-13 + 6 n) (-7 + 3 n) (-5 + 2 n) (-8 + 3 n) (-17 + 6 n) (-19 + 6 n) (-10 + 3 n) (-7 + 2 n) (-11 + 3 n))] and in Maple notation [-1/432*(-6+n)*(8140173088*n^12-111492692151*n^11+656024285066*n^10-\ 2194841122395*n^9+4726631213754*n^8-7237565748693*n^7+8700320858798*n^6-\ 8434277739705*n^5+5661505068958*n^4-1444641754656*n^3-1079176655664*n^2+ 987025435200*n-234897062400)/(-1+2*n)/(-5+6*n)/(-4+3*n)/(-11+6*n)/(-7+3*n)/(-2+ 3*n)/(-5+3*n)/(-8+3*n)/(-7+6*n)/(-13+6*n)/(-1+3*n)/(-1+6*n)/(-3+2*n), -1/7776*( 2104766143624*n^15-47627214026475*n^14+488795392979485*n^13-3025124682496785*n^ 12+12627976835537773*n^11-37586670240069555*n^10+82120994094584555*n^9-\ 133612606028709555*n^8+162798073490408807*n^7-148331514708096390*n^6+ 99814924682362460*n^5-47968841564051160*n^4+15573122735167296*n^3-\ 3417337339606080*n^2+594947511648000*n-22230464256000)/(-3+2*n)/(-1+6*n)/(-1+3* n)/(-13+6*n)/(-7+6*n)/(-8+3*n)/(-5+3*n)/(-2+3*n)/(-7+3*n)/(-11+6*n)/(-4+3*n)/(-\ 5+6*n)/(-17+6*n)/(-1+2*n)/(-5+2*n), -1/46656*(228495560674148*n^17-\ 6626881465806852*n^16+88579572156814815*n^15-723897701111039985*n^14+ 4045214584187791791*n^13-16376944780518197589*n^12+49649418272449484465*n^11-\ 114894364857385503555*n^10+204978474965297469249*n^9-282763634392819583091*n^8+ 300644189272431839100*n^7-244052759862715149360*n^6+148732057879868609312*n^5-\ 66254987497487989968*n^4+20693309813190933120*n^3-4233509772879513600*n^2+ 500597582002560000*n-25342729251840000)/(-3+2*n)/(-1+6*n)/(-1+3*n)/(-13+6*n)/(-\ 7+6*n)/(-10+3*n)/(-8+3*n)/(-5+3*n)/(-2+3*n)/(-7+3*n)/(-11+6*n)/(-19+6*n)/(-4+3* n)/(-5+6*n)/(-17+6*n)/(-1+2*n)/(-5+2*n), -1/186624*(5484236552343677*n^19-\ 198346606185984214*n^18+3336092223077677293*n^17-34650302010934018956*n^16+ 248878437911469111114*n^15-1311784196618066653188*n^14+5254342404401313571106*n ^13-16341263153211233064352*n^12+39973362649052989919961*n^11-\ 77430086862436662302982*n^10+118984069950522233477409*n^9-\ 144659298175675701727428*n^8+138145279824914851070528*n^7-\ 102340173735079242333616*n^6+57701738348286914046192*n^5-\ 24067533584280604443264*n^4+7113806966704179582720*n^3-1390568932571640192000*n ^2+158543004688104960000*n-7805560609566720000)/(-1+6*n)/(-1+3*n)/(-1+2*n)/(-2+ 3*n)/(-5+6*n)/(-7+6*n)/(-4+3*n)/(-3+2*n)/(-5+3*n)/(-11+6*n)/(-13+6*n)/(-7+3*n)/ (-5+2*n)/(-8+3*n)/(-17+6*n)/(-19+6*n)/(-10+3*n)/(-7+2*n)/(-11+3*n)] The limits, as n goes to infinity are -254380409 -263095767953 -57123890168537 -5484236552343677 [----------, -------------, ---------------, -----------------] 306110016 264479053824 57127475625984 5484237660094464 and in Maple notation [-254380409/306110016, -263095767953/264479053824, -57123890168537/ 57127475625984, -5484236552343677/5484237660094464] and in floating point [-.8310097537, -.9947697716, -.9999372376, -.9999997980] The cut off is at j=, 1 The rational functions describing the sorting probabilities of the cell, [1, 5], vs. those in the, 6, -th row from j=1 to j=, 4, are as follws 17 16 15 [- (75040883424751 n - 2199240972411804 n + 29546312803698855 n 14 13 - 241741939502957445 n + 1349407068105600357 n 12 11 - 5454094341711774633 n + 16523561472576845185 n 10 9 - 38281246067521594335 n + 68480515218618134808 n 8 7 - 94666382250316217847 n + 100354475368206480540 n 6 5 - 80333399909559936720 n + 47701663225432997584 n 4 3 - 21113180539591875216 n + 7719139526659729920 n 2 - 2852313565157568000 n + 952302880432512000 n - 166135669539840000)/( 15552 (-1 + 6 n) (-1 + 3 n) (-1 + 2 n) (-2 + 3 n) (-5 + 6 n) (-7 + 6 n) (-4 + 3 n) (-3 + 2 n) (-5 + 3 n) (-11 + 6 n) (-13 + 6 n) (-7 + 3 n) (-5 + 2 n) (-8 + 3 n) (-17 + 6 n) (-19 + 6 n) (-10 + 3 n)), - 5 ( 18 17 16 60932091455068 n - 1980442002057597 n + 29806344913749772 n 15 14 - 275713001041929158 n + 1754359989316708942 n 13 12 - 8142734486557516160 n + 28525028845224507398 n 11 10 - 76977758421333975770 n + 161895286203923165102 n 9 8 - 266718288250815414739 n + 344086818727405772438 n 7 6 - 345676061714129230336 n + 267436644558985930848 n 5 4 - 156498771037076450704 n + 67369744401682764192 n 3 2 - 20424603003953607936 n + 4075552330602762240 n - 484100790311577600 n + 23653213968384000)/(31104 (-7 + 2 n) (-1 + 6 n) (-1 + 3 n) (-1 + 2 n) (-2 + 3 n) (-5 + 6 n) (-7 + 6 n) (-4 + 3 n) (-3 + 2 n) (-5 + 3 n) (-11 + 6 n) (-13 + 6 n) (-7 + 3 n) (-5 + 2 n) (-8 + 3 n) (-17 + 6 n) 19 18 (-19 + 6 n) (-10 + 3 n)), - (2742118502336277 n - 99173303799129539 n 17 16 + 1668046082879649588 n - 17325151074928429506 n 15 14 + 124439219641612245054 n - 655892095100624186658 n 13 12 + 2627171198925045930036 n - 8170631619534721807232 n 11 10 + 19986681274489232305101 n - 38715043229780895928467 n 9 8 + 59492035549521151746864 n - 72329649034298778480918 n 7 6 + 69072638273155422260328 n - 51170089144865732955536 n 5 4 + 28850868840740536588512 n - 12033765109805551837344 n 3 2 + 3556904934452407338240 n - 695284086480102604800 n + 79271502344052480000 n - 3902780304783360000)/(93312 (-1 + 6 n) (-1 + 3 n) (-1 + 2 n) (-2 + 3 n) (-5 + 6 n) (-7 + 6 n) (-4 + 3 n) (-3 + 2 n) (-5 + 3 n) (-11 + 6 n) (-13 + 6 n) (-7 + 3 n) (-5 + 2 n) (-8 + 3 n) (-17 + 6 n) (-19 + 6 n) (-10 + 3 n) (-7 + 2 n) (-11 + 3 n)), - ( 20 19 197432555751714887 n - 7897302230886602670 n 18 17 + 147471150675324239245 n - 1707791607388122966090 n 16 15 + 13741365445886641783902 n - 81569455409209631545980 n 14 13 + 370182545769706630485290 n - 1313384725886888703309780 n 12 11 + 3694135369698642791771347 n - 8303807171160084094217190 n 10 9 + 14968778508280629485098185 n - 21627536388859524168307170 n 8 7 + 24936213216745297733354552 n - 22748294870768532017586000 n 6 5 + 16200206560069507974531280 n - 8829271099631083215756960 n 4 3 + 3577416684220997588685312 n - 1031765843599703056028160 n 2 + 197606060863658125056000 n - 22159934828902886400000 n + 1077167364120207360000)/(1119744 (-3 + 2 n) (-1 + 6 n) (-1 + 3 n) (-23 + 6 n) (-13 + 6 n) (-7 + 6 n) (-10 + 3 n) (-8 + 3 n) (-11 + 3 n) (-5 + 3 n) (-2 + 3 n) (-7 + 3 n) (-11 + 6 n) (-19 + 6 n) (-4 + 3 n) (-5 + 6 n) (-17 + 6 n) (-7 + 2 n) (-1 + 2 n) (-5 + 2 n))] and in Maple notation [-1/15552*(75040883424751*n^17-2199240972411804*n^16+29546312803698855*n^15-\ 241741939502957445*n^14+1349407068105600357*n^13-5454094341711774633*n^12+ 16523561472576845185*n^11-38281246067521594335*n^10+68480515218618134808*n^9-\ 94666382250316217847*n^8+100354475368206480540*n^7-80333399909559936720*n^6+ 47701663225432997584*n^5-21113180539591875216*n^4+7719139526659729920*n^3-\ 2852313565157568000*n^2+952302880432512000*n-166135669539840000)/(-1+6*n)/(-1+3 *n)/(-1+2*n)/(-2+3*n)/(-5+6*n)/(-7+6*n)/(-4+3*n)/(-3+2*n)/(-5+3*n)/(-11+6*n)/(-\ 13+6*n)/(-7+3*n)/(-5+2*n)/(-8+3*n)/(-17+6*n)/(-19+6*n)/(-10+3*n), -5/31104*( 60932091455068*n^18-1980442002057597*n^17+29806344913749772*n^16-\ 275713001041929158*n^15+1754359989316708942*n^14-8142734486557516160*n^13+ 28525028845224507398*n^12-76977758421333975770*n^11+161895286203923165102*n^10-\ 266718288250815414739*n^9+344086818727405772438*n^8-345676061714129230336*n^7+ 267436644558985930848*n^6-156498771037076450704*n^5+67369744401682764192*n^4-\ 20424603003953607936*n^3+4075552330602762240*n^2-484100790311577600*n+ 23653213968384000)/(-7+2*n)/(-1+6*n)/(-1+3*n)/(-1+2*n)/(-2+3*n)/(-5+6*n)/(-7+6* n)/(-4+3*n)/(-3+2*n)/(-5+3*n)/(-11+6*n)/(-13+6*n)/(-7+3*n)/(-5+2*n)/(-8+3*n)/(-\ 17+6*n)/(-19+6*n)/(-10+3*n), -1/93312*(2742118502336277*n^19-99173303799129539* n^18+1668046082879649588*n^17-17325151074928429506*n^16+124439219641612245054*n ^15-655892095100624186658*n^14+2627171198925045930036*n^13-\ 8170631619534721807232*n^12+19986681274489232305101*n^11-\ 38715043229780895928467*n^10+59492035549521151746864*n^9-\ 72329649034298778480918*n^8+69072638273155422260328*n^7-51170089144865732955536 *n^6+28850868840740536588512*n^5-12033765109805551837344*n^4+ 3556904934452407338240*n^3-695284086480102604800*n^2+79271502344052480000*n-\ 3902780304783360000)/(-1+6*n)/(-1+3*n)/(-1+2*n)/(-2+3*n)/(-5+6*n)/(-7+6*n)/(-4+ 3*n)/(-3+2*n)/(-5+3*n)/(-11+6*n)/(-13+6*n)/(-7+3*n)/(-5+2*n)/(-8+3*n)/(-17+6*n) /(-19+6*n)/(-10+3*n)/(-7+2*n)/(-11+3*n), -1/1119744*(197432555751714887*n^20-\ 7897302230886602670*n^19+147471150675324239245*n^18-1707791607388122966090*n^17 +13741365445886641783902*n^16-81569455409209631545980*n^15+ 370182545769706630485290*n^14-1313384725886888703309780*n^13+ 3694135369698642791771347*n^12-8303807171160084094217190*n^11+ 14968778508280629485098185*n^10-21627536388859524168307170*n^9+ 24936213216745297733354552*n^8-22748294870768532017586000*n^7+ 16200206560069507974531280*n^6-8829271099631083215756960*n^5+ 3577416684220997588685312*n^4-1031765843599703056028160*n^3+ 197606060863658125056000*n^2-22159934828902886400000*n+1077167364120207360000)/ (-3+2*n)/(-1+6*n)/(-1+3*n)/(-23+6*n)/(-13+6*n)/(-7+6*n)/(-10+3*n)/(-8+3*n)/(-11 +3*n)/(-5+3*n)/(-2+3*n)/(-7+3*n)/(-11+6*n)/(-19+6*n)/(-4+3*n)/(-5+6*n)/(-17+6*n )/(-7+2*n)/(-1+2*n)/(-5+2*n)] The limits, as n goes to infinity are -75040883424751 -76165114318835 -914039500778759 -197432555751714887 [---------------, ---------------, ----------------, -------------------] 76169967501312 76169967501312 914039610015744 197432555763400704 and in Maple notation [-75040883424751/76169967501312, -76165114318835/76169967501312, -\ 914039500778759/914039610015744, -197432555751714887/197432555763400704] and in floating point [-.9851767814, -.9999362848, -.9999998805, -.9999999999] The cut off is at j=, 1 ------------------------- This ends this article that took, 4624.614, seconds to produce ----------------------- This took, 4624.614, seconds.