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, n], and its Limiting behavior as n goes to infinity for i from 2 to, 4 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 -7 + n [- --------] -1 + 7 n and in Maple notation [-(-7+n)/(-1+7*n)] The limits, as n goes to infinity are [-1/7] and in Maple notation [-1/7] and in floating point [-.1428571429] 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 2 3 (13 n - 17 n - 6) [- ---------------------] (-2 + 7 n) (-1 + 7 n) and in Maple notation [-3*(13*n^2-17*n-6)/(-2+7*n)/(-1+7*n)] The limits, as n goes to infinity are -39 [---] 49 and in Maple notation [-39/49] and in floating point [-.7959183673] 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 3 (111 n - 118 n - 11 n - 22) [- --------------------------------] (-3 + 7 n) (-2 + 7 n) (-1 + 7 n) and in Maple notation [-3*(111*n^3-118*n^2-11*n-22)/(-3+7*n)/(-2+7*n)/(-1+7*n)] The limits, as n goes to infinity are -333 [----] 343 and in Maple notation [-333/343] and in floating point [-.9708454810] 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 5 (479 n - 698 n + 301 n - 130 n - 24) [- -------------------------------------------] (-4 + 7 n) (-3 + 7 n) (-2 + 7 n) (-1 + 7 n) and in Maple notation [-5*(479*n^4-698*n^3+301*n^2-130*n-24)/(-4+7*n)/(-3+7*n)/(-2+7*n)/(-1+7*n)] The limits, as n goes to infinity are -2395 [-----] 2401 and in Maple notation [-2395/2401] and in floating point [-.9975010412] 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 5 (3361 n - 7209 n + 5797 n - 2295 n + 274 n - 72) [- ------------------------------------------------------] (-5 + 7 n) (-4 + 7 n) (-3 + 7 n) (-2 + 7 n) (-1 + 7 n) and in Maple notation [-5*(3361*n^5-7209*n^4+5797*n^3-2295*n^2+274*n-72)/(-5+7*n)/(-4+7*n)/(-3+7*n)/( -2+7*n)/(-1+7*n)] The limits, as n goes to infinity are -16805 [------] 16807 and in Maple notation [-16805/16807] and in floating point [-.9998810020] The cut off is at j=, 1 The rational functions describing the sorting probabilities of the cell, [1, 2], vs. those in the, 7, -th row from j=1 to j=, 1, are as follws 6 5 4 3 2 [- (823541 n - 2470671 n + 2940875 n - 1766205 n + 553784 n - 89964 n + 3600)/(7 (-1 + 7 n) (-2 + 7 n) (-3 + 7 n) (-4 + 7 n) (-5 + 7 n) (-6 + 7 n))] and in Maple notation [-1/7*(823541*n^6-2470671*n^5+2940875*n^4-1766205*n^3+553784*n^2-89964*n+3600)/ (-1+7*n)/(-2+7*n)/(-3+7*n)/(-4+7*n)/(-5+7*n)/(-6+7*n)] The limits, as n goes to infinity are -823541 [-------] 823543 and in Maple notation [-823541/823543] and in floating point [-.9999975715] 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 25 n + 51 n - 46 6 5 4 3 [---------------------, - (373477 n - 2451681 n + 4905109 n - 4395231 n (-2 + 7 n) (-1 + 7 n) 2 + 1904062 n - 364680 n + 20160)/(7 (-8 + 7 n) (-1 + 7 n) (-3 + 7 n) (-4 + 7 n) (-5 + 7 n) (-6 + 7 n))] and in Maple notation [(25*n^2+51*n-46)/(-2+7*n)/(-1+7*n), -1/7*(373477*n^6-2451681*n^5+4905109*n^4-\ 4395231*n^3+1904062*n^2-364680*n+20160)/(-8+7*n)/(-1+7*n)/(-3+7*n)/(-4+7*n)/(-5 +7*n)/(-6+7*n)] The limits, as n goes to infinity are 25 -373477 [--, -------] 49 823543 and in Maple notation [25/49, -373477/823543] and in floating point [.5102040816, -.4535003030] 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 2 (-7 + n) (113 n - 42 n - 56) 8 7 [- --------------------------------, - (37856941 n - 222177458 n (-2 + 7 n) (-4 + 7 n) (-1 + 7 n) 6 5 4 3 2 + 505441660 n - 609824786 n + 439793539 n - 198074972 n + 54304740 n - 7836624 n + 362880)/(7 (-9 + 7 n) (-8 + 7 n) (-6 + 7 n) (-5 + 7 n) (-4 + 7 n) (-3 + 7 n) (-2 + 7 n) (-1 + 7 n))] and in Maple notation [-(-7+n)*(113*n^2-42*n-56)/(-2+7*n)/(-4+7*n)/(-1+7*n), -1/7*(37856941*n^8-\ 222177458*n^7+505441660*n^6-609824786*n^5+439793539*n^4-198074972*n^3+54304740* n^2-7836624*n+362880)/(-9+7*n)/(-8+7*n)/(-6+7*n)/(-5+7*n)/(-4+7*n)/(-3+7*n)/(-2 +7*n)/(-1+7*n)] The limits, as n goes to infinity are -113 -37856941 [----, ---------] 343 40353607 and in Maple notation [-113/343, -37856941/40353607] and in floating point [-.3294460641, -.9381302891] 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 6 5 4 3 2 97811 n - 429513 n + 427085 n - 74955 n + 50024 n - 113652 n + 42480 [- -------------------------------------------------------------------------, - (-6 + 7 n) (-5 + 7 n) (-4 + 7 n) (-3 + 7 n) (-2 + 7 n) (-1 + 7 n) 9 8 7 6 (281589979 n - 1942758336 n + 5662973784 n - 9267838806 n 5 4 3 2 + 9311951241 n - 5873154114 n + 2309793716 n - 556158744 n + 76603680 n - 3628800)/(7 (-10 + 7 n) (-9 + 7 n) (-8 + 7 n) (-6 + 7 n) (-5 + 7 n) (-4 + 7 n) (-3 + 7 n) (-2 + 7 n) (-1 + 7 n))] and in Maple notation [-(97811*n^6-429513*n^5+427085*n^4-74955*n^3+50024*n^2-113652*n+42480)/(-6+7*n) /(-5+7*n)/(-4+7*n)/(-3+7*n)/(-2+7*n)/(-1+7*n), -1/7*(281589979*n^9-1942758336*n ^8+5662973784*n^7-9267838806*n^6+9311951241*n^5-5873154114*n^4+2309793716*n^3-\ 556158744*n^2+76603680*n-3628800)/(-10+7*n)/(-9+7*n)/(-8+7*n)/(-6+7*n)/(-5+7*n) /(-4+7*n)/(-3+7*n)/(-2+7*n)/(-1+7*n)] The limits, as n goes to infinity are -13973 -281589979 [------, ----------] 16807 282475249 and in Maple notation [-13973/16807, -281589979/282475249] and in floating point [-.8313797822, -.9968660263] 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 [- (803957 n - 3563959 n + 5482259 n - 4908565 n + 3182288 n - 1067116 n - 262224 n + 213120)/((-8 + 7 n) (-6 + 7 n) (-5 + 7 n) (-4 + 7 n) 10 9 (-3 + 7 n) (-2 + 7 n) (-1 + 7 n)), - 5 (395438069 n - 3333591265 n 8 7 6 5 + 12200863464 n - 25469318610 n + 33402611721 n - 28648417641 n 4 3 2 + 16197325618 n - 5899946948 n + 1311205032 n - 164522592 n + 7983360)/ (7 (-11 + 7 n) (-10 + 7 n) (-9 + 7 n) (-8 + 7 n) (-6 + 7 n) (-5 + 7 n) (-4 + 7 n) (-3 + 7 n) (-2 + 7 n) (-1 + 7 n))] and in Maple notation [-(803957*n^7-3563959*n^6+5482259*n^5-4908565*n^4+3182288*n^3-1067116*n^2-\ 262224*n+213120)/(-8+7*n)/(-6+7*n)/(-5+7*n)/(-4+7*n)/(-3+7*n)/(-2+7*n)/(-1+7*n) , -5/7*(395438069*n^10-3333591265*n^9+12200863464*n^8-25469318610*n^7+ 33402611721*n^6-28648417641*n^5+16197325618*n^4-5899946948*n^3+1311205032*n^2-\ 164522592*n+7983360)/(-11+7*n)/(-10+7*n)/(-9+7*n)/(-8+7*n)/(-6+7*n)/(-5+7*n)/(-\ 4+7*n)/(-3+7*n)/(-2+7*n)/(-1+7*n)] The limits, as n goes to infinity are -114851 -1977190345 [-------, -----------] 117649 1977326743 and in Maple notation [-114851/117649, -1977190345/1977326743] and in floating point [-.9762173924, -.9999310190] 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 [- (281978123 n - 1942616256 n + 5654943672 n - 9269338614 n 5 4 3 2 + 9364846617 n - 5880325794 n + 2181452308 n - 500385816 n + 158205600 n - 52358400)/(7 (-10 + 7 n) (-9 + 7 n) (-8 + 7 n) (-6 + 7 n) (-5 + 7 n) (-4 + 7 n) (-3 + 7 n) (-2 + 7 n) (-1 + 7 n)), - 5 ( 11 10 9 8 2768255843 n - 28078077859 n + 125418624852 n - 324687265470 n 7 6 5 4 + 539408874855 n - 601473382539 n + 457140328858 n - 235526715860 n 3 2 + 80079714312 n - 16925755392 n + 1990465920 n - 95800320)/(7 (-12 + 7 n) (-11 + 7 n) (-10 + 7 n) (-9 + 7 n) (-8 + 7 n) (-6 + 7 n) (-5 + 7 n) (-4 + 7 n) (-3 + 7 n) (-2 + 7 n) (-1 + 7 n))] and in Maple notation [-1/7*(281978123*n^9-1942616256*n^8+5654943672*n^7-9269338614*n^6+9364846617*n^ 5-5880325794*n^4+2181452308*n^3-500385816*n^2+158205600*n-52358400)/(-10+7*n)/( -9+7*n)/(-8+7*n)/(-6+7*n)/(-5+7*n)/(-4+7*n)/(-3+7*n)/(-2+7*n)/(-1+7*n), -5/7*( 2768255843*n^11-28078077859*n^10+125418624852*n^9-324687265470*n^8+539408874855 *n^7-601473382539*n^6+457140328858*n^5-235526715860*n^4+80079714312*n^3-\ 16925755392*n^2+1990465920*n-95800320)/(-12+7*n)/(-11+7*n)/(-10+7*n)/(-9+7*n)/( -8+7*n)/(-6+7*n)/(-5+7*n)/(-4+7*n)/(-3+7*n)/(-2+7*n)/(-1+7*n)] The limits, as n goes to infinity are -40282589 -13841279215 [---------, ------------] 40353607 13841287201 and in Maple notation [-40282589/40353607, -13841279215/13841287201] and in floating point [-.9982401078, -.9999994230] The cut off is at j=, 1 The rational functions describing the sorting probabilities of the cell, [1, 3], vs. those in the, 7, -th row from j=1 to j=, 2, are as follws 11 10 9 8 [- (13840549235 n - 140401688899 n + 627041180100 n - 1623443203710 n 7 6 5 + 2697476990295 n - 3006988483947 n + 2284255799530 n 4 3 2 - 1178165397140 n + 403222990920 n - 85463080704 n + 8145947520 n + 479001600)/(7 (-12 + 7 n) (-11 + 7 n) (-10 + 7 n) (-9 + 7 n) (-8 + 7 n) (-6 + 7 n) (-5 + 7 n) (-4 + 7 n) (-3 + 7 n) (-2 + 7 n) (-1 + 7 n)), - ( 12 11 10 678223071991 n - 8138676905076 n + 43503165179393 n 9 8 7 - 136613509281330 n + 279888595240173 n - 392792117685048 n 6 5 4 + 385666064789639 n - 265709947546410 n + 126780883231636 n 3 2 - 40578114185976 n + 8195577015168 n - 926820178560 n + 43589145600)/(49 (-13 + 7 n) (-12 + 7 n) (-11 + 7 n) (-10 + 7 n) (-9 + 7 n) (-8 + 7 n) (-6 + 7 n) (-5 + 7 n) (-4 + 7 n) (-3 + 7 n) (-2 + 7 n) (-1 + 7 n))] and in Maple notation [-1/7*(13840549235*n^11-140401688899*n^10+627041180100*n^9-1623443203710*n^8+ 2697476990295*n^7-3006988483947*n^6+2284255799530*n^5-1178165397140*n^4+ 403222990920*n^3-85463080704*n^2+8145947520*n+479001600)/(-12+7*n)/(-11+7*n)/(-\ 10+7*n)/(-9+7*n)/(-8+7*n)/(-6+7*n)/(-5+7*n)/(-4+7*n)/(-3+7*n)/(-2+7*n)/(-1+7*n) , -1/49*(678223071991*n^12-8138676905076*n^11+43503165179393*n^10-\ 136613509281330*n^9+279888595240173*n^8-392792117685048*n^7+385666064789639*n^6 -265709947546410*n^5+126780883231636*n^4-40578114185976*n^3+8195577015168*n^2-\ 926820178560*n+43589145600)/(-13+7*n)/(-12+7*n)/(-11+7*n)/(-10+7*n)/(-9+7*n)/(-\ 8+7*n)/(-6+7*n)/(-5+7*n)/(-4+7*n)/(-3+7*n)/(-2+7*n)/(-1+7*n)] The limits, as n goes to infinity are -13840549235 -678223071991 [------------, -------------] 13841287201 678223072849 and in Maple notation [-13840549235/13841287201, -678223071991/678223072849] and in floating point [-.9999466837, -.9999999987] 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 2 283 n + 66 n - 583 n + 354 8 7 6 [--------------------------------, (7422877 n + 45228862 n - 317112788 n (-3 + 7 n) (-1 + 7 n) (-2 + 7 n) 5 4 3 2 + 647580382 n - 636324317 n + 333478468 n - 91623132 n + 11385648 n - 362880)/(7 (-1 + 7 n) (-2 + 7 n) (-3 + 7 n) (-4 + 7 n) (-5 + 7 n) 12 11 (-6 + 7 n) (-8 + 7 n) (-9 + 7 n)), - (411255228523 n - 6595641278173 n 10 9 8 + 45742393707003 n - 182808079869717 n + 470571235374237 n 7 6 5 - 822736580013411 n + 1000801638127817 n - 850946835030023 n 4 3 2 + 499331278674900 n - 195514644841956 n + 47778384198240 n - 6361363512000 n + 326918592000)/(49 (-15 + 7 n) (-13 + 7 n) (-12 + 7 n) (-11 + 7 n) (-10 + 7 n) (-9 + 7 n) (-8 + 7 n) (-6 + 7 n) (-5 + 7 n) (-4 + 7 n) (-3 + 7 n) (-1 + 7 n))] and in Maple notation [(283*n^3+66*n^2-583*n+354)/(-3+7*n)/(-1+7*n)/(-2+7*n), 1/7*(7422877*n^8+ 45228862*n^7-317112788*n^6+647580382*n^5-636324317*n^4+333478468*n^3-91623132*n ^2+11385648*n-362880)/(-1+7*n)/(-2+7*n)/(-3+7*n)/(-4+7*n)/(-5+7*n)/(-6+7*n)/(-8 +7*n)/(-9+7*n), -1/49*(411255228523*n^12-6595641278173*n^11+45742393707003*n^10 -182808079869717*n^9+470571235374237*n^8-822736580013411*n^7+1000801638127817*n ^6-850946835030023*n^5+499331278674900*n^4-195514644841956*n^3+47778384198240*n ^2-6361363512000*n+326918592000)/(-15+7*n)/(-13+7*n)/(-12+7*n)/(-11+7*n)/(-10+7 *n)/(-9+7*n)/(-8+7*n)/(-6+7*n)/(-5+7*n)/(-4+7*n)/(-3+7*n)/(-1+7*n)] The limits, as n goes to infinity are 283 1060411 -411255228523 [---, -------, -------------] 343 5764801 678223072849 and in Maple notation [283/343, 1060411/5764801, -411255228523/678223072849] and in floating point [.8250728863, .1839458118, -.6063716275] 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 6 5 4 3 2 [(21553 n + 288141 n - 985025 n + 781815 n + 233032 n - 451836 n + 113040) /((-1 + 7 n) (-2 + 7 n) (-3 + 7 n) (-4 + 7 n) (-5 + 7 n) (-6 + 7 n)), - ( 10 9 8 7 1380476027 n - 15356900479 n + 63680904216 n - 135003298734 n 6 5 4 3 + 167076461127 n - 132687995991 n + 75183464974 n - 32917666076 n 2 + 10211541336 n - 1617495840 n + 39916800)/(7 (-1 + 7 n) (-2 + 7 n) (-3 + 7 n) (-4 + 7 n) (-5 + 7 n) (-6 + 7 n) (-8 + 7 n) (-9 + 7 n) 14 13 (-10 + 7 n) (-11 + 7 n)), - 5 (6474495856073 n - 108072038418029 n 12 11 10 + 810206898488635 n - 3620338289933941 n + 10773336500957793 n 9 8 7 - 22566266315224131 n + 34252215074978353 n - 38171334490878631 n 6 5 4 + 31273798904739170 n - 18660551786347700 n + 7936714203174792 n 3 2 - 2316534659987328 n + 434635168581504 n - 46375579192320 n + 2092278988800)/(49 (-16 + 7 n) (-15 + 7 n) (-13 + 7 n) (-12 + 7 n) (-1 + 7 n) (-2 + 7 n) (-3 + 7 n) (-4 + 7 n) (-5 + 7 n) (-6 + 7 n) (-8 + 7 n) (-9 + 7 n) (-10 + 7 n) (-11 + 7 n))] and in Maple notation [(21553*n^6+288141*n^5-985025*n^4+781815*n^3+233032*n^2-451836*n+113040)/(-1+7* n)/(-2+7*n)/(-3+7*n)/(-4+7*n)/(-5+7*n)/(-6+7*n), -1/7*(1380476027*n^10-\ 15356900479*n^9+63680904216*n^8-135003298734*n^7+167076461127*n^6-132687995991* n^5+75183464974*n^4-32917666076*n^3+10211541336*n^2-1617495840*n+39916800)/(-1+ 7*n)/(-2+7*n)/(-3+7*n)/(-4+7*n)/(-5+7*n)/(-6+7*n)/(-8+7*n)/(-9+7*n)/(-10+7*n)/( -11+7*n), -5/49*(6474495856073*n^14-108072038418029*n^13+810206898488635*n^12-\ 3620338289933941*n^11+10773336500957793*n^10-22566266315224131*n^9+ 34252215074978353*n^8-38171334490878631*n^7+31273798904739170*n^6-\ 18660551786347700*n^5+7936714203174792*n^4-2316534659987328*n^3+434635168581504 *n^2-46375579192320*n+2092278988800)/(-16+7*n)/(-15+7*n)/(-13+7*n)/(-12+7*n)/(-\ 1+7*n)/(-2+7*n)/(-3+7*n)/(-4+7*n)/(-5+7*n)/(-6+7*n)/(-8+7*n)/(-9+7*n)/(-10+7*n) /(-11+7*n)] The limits, as n goes to infinity are 3079 -197210861 -4624639897195 [-----, ----------, --------------] 16807 282475249 4747561509943 and in Maple notation [3079/16807, -197210861/282475249, -4624639897195/4747561509943] and in floating point [.1831974772, -.6981527114, -.9741084739] 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 6 5 4 3 2 [- (-7 + n) (447209 n - 1197427 n + 412135 n + 548695 n + 218376 n - 556668 n + 187920)/((-9 + 7 n) (-1 + 7 n) (-2 + 7 n) (-3 + 7 n) 12 11 (-4 + 7 n) (-5 + 7 n) (-6 + 7 n)), - (93750782531 n - 1166856258372 n 10 9 8 + 6258776246917 n - 19484544343530 n + 39748212093873 n 7 6 5 - 56160991456056 n + 55672142320771 n - 38058395707650 n 4 3 2 + 17506658968196 n - 5557304043672 n + 1394128691712 n - 251338273920 n + 6227020800)/(7 (-13 + 7 n) (-12 + 7 n) (-11 + 7 n) (-10 + 7 n) (-9 + 7 n) (-8 + 7 n) (-6 + 7 n) (-5 + 7 n) (-4 + 7 n) (-3 + 7 n) 15 14 (-2 + 7 n) (-1 + 7 n)), - 5 (46497626196647 n - 877378248864084 n 13 12 11 + 7541482847838170 n - 39128082598384638 n + 136815397907762972 n 10 9 8 - 340799667331561830 n + 623255426303593450 n - 849815523121731954 n 7 6 5 + 868321519738130605 n - 662440185950761038 n + 372615444040462540 n 4 3 2 - 150948662522982408 n + 42348559136875776 n - 7702293540786048 n + 803030799191040 n - 35568742809600)/(49 (-2 + 7 n) (-1 + 7 n) (-3 + 7 n) (-6 + 7 n) (-8 + 7 n) (-5 + 7 n) (-9 + 7 n) (-4 + 7 n) (-17 + 7 n) (-11 + 7 n) (-15 + 7 n) (-13 + 7 n) (-16 + 7 n) (-10 + 7 n) (-12 + 7 n))] and in Maple notation [-(-7+n)*(447209*n^6-1197427*n^5+412135*n^4+548695*n^3+218376*n^2-556668*n+ 187920)/(-9+7*n)/(-1+7*n)/(-2+7*n)/(-3+7*n)/(-4+7*n)/(-5+7*n)/(-6+7*n), -1/7*( 93750782531*n^12-1166856258372*n^11+6258776246917*n^10-19484544343530*n^9+ 39748212093873*n^8-56160991456056*n^7+55672142320771*n^6-38058395707650*n^5+ 17506658968196*n^4-5557304043672*n^3+1394128691712*n^2-251338273920*n+ 6227020800)/(-13+7*n)/(-12+7*n)/(-11+7*n)/(-10+7*n)/(-9+7*n)/(-8+7*n)/(-6+7*n)/ (-5+7*n)/(-4+7*n)/(-3+7*n)/(-2+7*n)/(-1+7*n), -5/49*(46497626196647*n^15-\ 877378248864084*n^14+7541482847838170*n^13-39128082598384638*n^12+ 136815397907762972*n^11-340799667331561830*n^10+623255426303593450*n^9-\ 849815523121731954*n^8+868321519738130605*n^7-662440185950761038*n^6+ 372615444040462540*n^5-150948662522982408*n^4+42348559136875776*n^3-\ 7702293540786048*n^2+803030799191040*n-35568742809600)/(-2+7*n)/(-1+7*n)/(-3+7* n)/(-6+7*n)/(-8+7*n)/(-5+7*n)/(-9+7*n)/(-4+7*n)/(-17+7*n)/(-11+7*n)/(-15+7*n)/( -13+7*n)/(-16+7*n)/(-10+7*n)/(-12+7*n)] The limits, as n goes to infinity are -63887 -13392968933 -232488130983235 [------, ------------, ----------------] 117649 13841287201 232630513987207 and in Maple notation [-63887/117649, -13392968933/13841287201, -232488130983235/232630513987207] and in floating point [-.5430305400, -.9676100740, -.9993879436] 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 11 10 9 8 [- 5 (2498629903 n - 28555556687 n + 130522080948 n - 322655071158 n 7 6 5 4 + 503712420099 n - 576807693591 n + 536235412370 n - 357516555172 n 3 2 + 92433969768 n + 65183182848 n - 58762585728 n + 13398359040)/(7 (-12 + 7 n) (-11 + 7 n) (-10 + 7 n) (-9 + 7 n) (-8 + 7 n) (-6 + 7 n) (-5 + 7 n) (-4 + 7 n) (-3 + 7 n) (-2 + 7 n) (-1 + 7 n)), - ( 13 12 11 677264204477 n - 9600587830449 n + 60930805577239 n 10 9 8 - 229755449483865 n + 572829496979061 n - 992843928103167 n 7 6 5 + 1226422164656197 n - 1091616174905715 n + 698191143228362 n 4 3 2 - 312718017217284 n + 93165264852264 n - 18319748123520 n + 2741726102400 n - 93405312000)/(7 (-15 + 7 n) (-13 + 7 n) (-12 + 7 n) (-11 + 7 n) (-10 + 7 n) (-9 + 7 n) (-8 + 7 n) (-6 + 7 n) (-5 + 7 n) 16 (-4 + 7 n) (-3 + 7 n) (-2 + 7 n) (-1 + 7 n)), - (1628405553232093 n 15 14 13 - 34894673743801650 n + 342897037759118326 n - 2048183264511374718 n 12 11 + 8310189640065010000 n - 24241609890780556842 n 10 9 + 52485574511368936058 n - 85835766719873862714 n 8 7 + 106874973733230123971 n - 101335527436116581916 n 6 5 + 72661232165078021456 n - 38817732415409394768 n 4 3 2 + 15067290956547264336 n - 4081187849669361792 n + 721432492756565760 n - 73517677925529600 n + 3201186852864000)/(49 (-2 + 7 n) (-1 + 7 n) (-3 + 7 n) (-18 + 7 n) (-6 + 7 n) (-8 + 7 n) (-5 + 7 n) (-9 + 7 n) (-4 + 7 n) (-17 + 7 n) (-11 + 7 n) (-15 + 7 n) (-13 + 7 n) (-16 + 7 n) (-10 + 7 n) (-12 + 7 n))] and in Maple notation [-5/7*(2498629903*n^11-28555556687*n^10+130522080948*n^9-322655071158*n^8+ 503712420099*n^7-576807693591*n^6+536235412370*n^5-357516555172*n^4+92433969768 *n^3+65183182848*n^2-58762585728*n+13398359040)/(-12+7*n)/(-11+7*n)/(-10+7*n)/( -9+7*n)/(-8+7*n)/(-6+7*n)/(-5+7*n)/(-4+7*n)/(-3+7*n)/(-2+7*n)/(-1+7*n), -1/7*( 677264204477*n^13-9600587830449*n^12+60930805577239*n^11-229755449483865*n^10+ 572829496979061*n^9-992843928103167*n^8+1226422164656197*n^7-1091616174905715*n ^6+698191143228362*n^5-312718017217284*n^4+93165264852264*n^3-18319748123520*n^ 2+2741726102400*n-93405312000)/(-15+7*n)/(-13+7*n)/(-12+7*n)/(-11+7*n)/(-10+7*n )/(-9+7*n)/(-8+7*n)/(-6+7*n)/(-5+7*n)/(-4+7*n)/(-3+7*n)/(-2+7*n)/(-1+7*n), -1/ 49*(1628405553232093*n^16-34894673743801650*n^15+342897037759118326*n^14-\ 2048183264511374718*n^13+8310189640065010000*n^12-24241609890780556842*n^11+ 52485574511368936058*n^10-85835766719873862714*n^9+106874973733230123971*n^8-\ 101335527436116581916*n^7+72661232165078021456*n^6-38817732415409394768*n^5+ 15067290956547264336*n^4-4081187849669361792*n^3+721432492756565760*n^2-\ 73517677925529600*n+3201186852864000)/(-2+7*n)/(-1+7*n)/(-3+7*n)/(-18+7*n)/(-6+ 7*n)/(-8+7*n)/(-5+7*n)/(-9+7*n)/(-4+7*n)/(-17+7*n)/(-11+7*n)/(-15+7*n)/(-13+7*n )/(-16+7*n)/(-10+7*n)/(-12+7*n)] The limits, as n goes to infinity are -254962235 -96752029211 -1628405553232093 [----------, ------------, -----------------] 282475249 96889010407 1628413597910449 and in Maple notation [-254962235/282475249, -96752029211/96889010407, -1628405553232093/ 1628413597910449] and in floating point [-.9026002664, -.9985862050, -.9999950598] 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 [- (671081358101 n - 9625597009497 n + 61111940583247 n 10 9 8 - 229277412417825 n + 570462673168893 n - 994894094960631 n 7 6 5 + 1241056666180141 n - 1097866124272395 n + 666594227413706 n 4 3 2 - 271587438992772 n + 91962533334312 n - 46771915898880 n + 23072989449600 n - 4950481536000)/(7 (-1 + 7 n) (-2 + 7 n) (-3 + 7 n) (-4 + 7 n) (-5 + 7 n) (-6 + 7 n) (-8 + 7 n) (-9 + 7 n) (-10 + 7 n) 15 (-11 + 7 n) (-12 + 7 n) (-13 + 7 n) (-15 + 7 n)), - (232625369267627 n 14 13 12 - 4386832478825412 n + 37704674594908706 n - 195641832489117558 n 11 10 + 684096613576292828 n - 1703983414103804286 n 9 8 7 + 3116213067410193298 n - 4249166450610342714 n + 4341687786405374953 n 6 5 4 - 3311914047226537062 n + 1863136873272562396 n - 755194025624857128 n 3 2 + 211477614462369792 n - 38249199524499840 n + 4210701426662400 n - 177843714048000)/(49 (-2 + 7 n) (-1 + 7 n) (-3 + 7 n) (-6 + 7 n) (-8 + 7 n) (-5 + 7 n) (-9 + 7 n) (-4 + 7 n) (-17 + 7 n) (-11 + 7 n) (-15 + 7 n) (-13 + 7 n) (-16 + 7 n) (-10 + 7 n) (-12 + 7 n)), - ( 17 16 15 11398895054653411 n - 275201901528631597 n + 3063278568747898672 n 14 13 - 20852334857393038420 n + 97086786989838390682 n 12 11 - 327584836660082572294 n + 827989781757020493524 n 10 9 - 1598076290489483053700 n + 2379003833282014878203 n 8 7 - 2739973448387878816421 n + 2434004517556105947236 n 6 5 - 1652286943091962943840 n + 843007261478774438304 n 4 3 - 314847389096318120688 n + 82592816820944957568 n 2 - 14221654916373239040 n + 1419244188555110400 n - 60822550204416000)/(49 (-1 + 7 n) (-2 + 7 n) (-3 + 7 n) (-4 + 7 n) (-5 + 7 n) (-6 + 7 n) (-8 + 7 n) (-9 + 7 n) (-10 + 7 n) (-11 + 7 n) (-12 + 7 n) (-13 + 7 n) (-15 + 7 n) (-16 + 7 n) (-17 + 7 n) (-18 + 7 n) (-19 + 7 n))] and in Maple notation [-1/7*(671081358101*n^13-9625597009497*n^12+61111940583247*n^11-229277412417825 *n^10+570462673168893*n^9-994894094960631*n^8+1241056666180141*n^7-\ 1097866124272395*n^6+666594227413706*n^5-271587438992772*n^4+91962533334312*n^3 -46771915898880*n^2+23072989449600*n-4950481536000)/(-1+7*n)/(-2+7*n)/(-3+7*n)/ (-4+7*n)/(-5+7*n)/(-6+7*n)/(-8+7*n)/(-9+7*n)/(-10+7*n)/(-11+7*n)/(-12+7*n)/(-13 +7*n)/(-15+7*n), -1/49*(232625369267627*n^15-4386832478825412*n^14+ 37704674594908706*n^13-195641832489117558*n^12+684096613576292828*n^11-\ 1703983414103804286*n^10+3116213067410193298*n^9-4249166450610342714*n^8+ 4341687786405374953*n^7-3311914047226537062*n^6+1863136873272562396*n^5-\ 755194025624857128*n^4+211477614462369792*n^3-38249199524499840*n^2+ 4210701426662400*n-177843714048000)/(-2+7*n)/(-1+7*n)/(-3+7*n)/(-6+7*n)/(-8+7*n )/(-5+7*n)/(-9+7*n)/(-4+7*n)/(-17+7*n)/(-11+7*n)/(-15+7*n)/(-13+7*n)/(-16+7*n)/ (-10+7*n)/(-12+7*n), -1/49*(11398895054653411*n^17-275201901528631597*n^16+ 3063278568747898672*n^15-20852334857393038420*n^14+97086786989838390682*n^13-\ 327584836660082572294*n^12+827989781757020493524*n^11-1598076290489483053700*n^ 10+2379003833282014878203*n^9-2739973448387878816421*n^8+2434004517556105947236 *n^7-1652286943091962943840*n^6+843007261478774438304*n^5-314847389096318120688 *n^4+82592816820944957568*n^3-14221654916373239040*n^2+1419244188555110400*n-\ 60822550204416000)/(-1+7*n)/(-2+7*n)/(-3+7*n)/(-4+7*n)/(-5+7*n)/(-6+7*n)/(-8+7* n)/(-9+7*n)/(-10+7*n)/(-11+7*n)/(-12+7*n)/(-13+7*n)/(-15+7*n)/(-16+7*n)/(-17+7* n)/(-18+7*n)/(-19+7*n)] The limits, as n goes to infinity are -95868765443 -33232195609661 -11398895054653411 [------------, ---------------, ------------------] 96889010407 33232930569601 11398895185373143 and in Maple notation [-95868765443/96889010407, -33232195609661/33232930569601, -11398895054653411/ 11398895185373143] and in floating point [-.9894699620, -.9999778846, -.9999999885] The cut off is at j=, 1 The rational functions describing the sorting probabilities of the cell, [1, 4], vs. those in the, 7, -th row from j=1 to j=, 3, are as follws 16 15 14 [- (1627679928918251 n - 34898719302102990 n + 342928488586286522 n 13 12 - 2048060377776150546 n + 8309476561567979600 n 11 10 - 24242641420815019494 n + 52494212807525479606 n 9 8 - 85839218019329351478 n + 106830020078793795157 n 7 6 - 101259195128784583572 n + 72702183199258917712 n 5 4 - 39046449250399777776 n + 15270098496185410992 n 3 2 - 4041114559276546944 n + 540281096594300160 n + 43157518912972800 n - 22408307970048000)/(49 (-18 + 7 n) (-17 + 7 n) (-16 + 7 n) (-15 + 7 n) (-13 + 7 n) (-12 + 7 n) (-11 + 7 n) (-10 + 7 n) (-9 + 7 n) (-8 + 7 n) (-6 + 7 n) (-5 + 7 n) (-4 + 7 n) (-3 + 7 n) (-2 + 7 n) (-1 + 7 n)), - ( 17 16 15 11398894202880203 n - 275201924034487541 n + 3063278344761251216 n 14 13 - 20852335730482215380 n + 97086787879166974826 n 12 11 - 327584818304846864342 n + 827989823269564654612 n 10 9 - 1598076370557135861700 n + 2379003404655367879939 n 8 7 - 2739973575304578626893 n + 2434006195430492327428 n 6 5 - 1652285118346511592160 n + 843004324433544361632 n 4 3 - 314852672504252890224 n + 82593774978305007744 n 2 - 14216280599012117760 n + 1421657781817190400 n - 60822550204416000)/(49 (-2 + 7 n) (-1 + 7 n) (-3 + 7 n) (-18 + 7 n) (-6 + 7 n) (-8 + 7 n) (-5 + 7 n) (-19 + 7 n) (-9 + 7 n) (-4 + 7 n) (-17 + 7 n) (-11 + 7 n) (-15 + 7 n) (-13 + 7 n) (-16 + 7 n) (-10 + 7 n) (-12 + 7 n)), - ( 18 17 558545864080512667 n - 15080738330373378489 n 16 15 + 188628917524994052204 n - 1450623401322345734580 n 14 13 + 7676579448669817877274 n - 29643807236124224111598 n 12 11 + 86433376168464247647868 n - 194224307993894421502740 n 10 9 + 340301869193911900326051 n - 467319233117443627400337 n 8 7 + 502862506072106977865952 n - 421722695431702430372880 n 6 5 + 272627521882953347711008 n - 133448541043221975726576 n 4 3 + 48125685256018149398976 n - 12259852457400834220800 n 2 + 2060575487945124480000 n - 201674491357731840000 n + 8515157028618240000)/(343 (-2 + 7 n) (-1 + 7 n) (-3 + 7 n) (-18 + 7 n) (-6 + 7 n) (-8 + 7 n) (-5 + 7 n) (-20 + 7 n) (-19 + 7 n) (-9 + 7 n) (-4 + 7 n) (-17 + 7 n) (-11 + 7 n) (-15 + 7 n) (-13 + 7 n) (-16 + 7 n) (-10 + 7 n) (-12 + 7 n))] and in Maple notation [-1/49*(1627679928918251*n^16-34898719302102990*n^15+342928488586286522*n^14-\ 2048060377776150546*n^13+8309476561567979600*n^12-24242641420815019494*n^11+ 52494212807525479606*n^10-85839218019329351478*n^9+106830020078793795157*n^8-\ 101259195128784583572*n^7+72702183199258917712*n^6-39046449250399777776*n^5+ 15270098496185410992*n^4-4041114559276546944*n^3+540281096594300160*n^2+ 43157518912972800*n-22408307970048000)/(-18+7*n)/(-17+7*n)/(-16+7*n)/(-15+7*n)/ (-13+7*n)/(-12+7*n)/(-11+7*n)/(-10+7*n)/(-9+7*n)/(-8+7*n)/(-6+7*n)/(-5+7*n)/(-4 +7*n)/(-3+7*n)/(-2+7*n)/(-1+7*n), -1/49*(11398894202880203*n^17-\ 275201924034487541*n^16+3063278344761251216*n^15-20852335730482215380*n^14+ 97086787879166974826*n^13-327584818304846864342*n^12+827989823269564654612*n^11 -1598076370557135861700*n^10+2379003404655367879939*n^9-2739973575304578626893* n^8+2434006195430492327428*n^7-1652285118346511592160*n^6+843004324433544361632 *n^5-314852672504252890224*n^4+82593774978305007744*n^3-14216280599012117760*n^ 2+1421657781817190400*n-60822550204416000)/(-2+7*n)/(-1+7*n)/(-3+7*n)/(-18+7*n) /(-6+7*n)/(-8+7*n)/(-5+7*n)/(-19+7*n)/(-9+7*n)/(-4+7*n)/(-17+7*n)/(-11+7*n)/(-\ 15+7*n)/(-13+7*n)/(-16+7*n)/(-10+7*n)/(-12+7*n), -1/343*(558545864080512667*n^ 18-15080738330373378489*n^17+188628917524994052204*n^16-1450623401322345734580* n^15+7676579448669817877274*n^14-29643807236124224111598*n^13+ 86433376168464247647868*n^12-194224307993894421502740*n^11+ 340301869193911900326051*n^10-467319233117443627400337*n^9+ 502862506072106977865952*n^8-421722695431702430372880*n^7+ 272627521882953347711008*n^6-133448541043221975726576*n^5+ 48125685256018149398976*n^4-12259852457400834220800*n^3+2060575487945124480000* n^2-201674491357731840000*n+8515157028618240000)/(-2+7*n)/(-1+7*n)/(-3+7*n)/(-\ 18+7*n)/(-6+7*n)/(-8+7*n)/(-5+7*n)/(-20+7*n)/(-19+7*n)/(-9+7*n)/(-4+7*n)/(-17+7 *n)/(-11+7*n)/(-15+7*n)/(-13+7*n)/(-16+7*n)/(-10+7*n)/(-12+7*n)] The limits, as n goes to infinity are -1627679928918251 -11398894202880203 -558545864080512667 [-----------------, ------------------, -------------------] 1628413597910449 11398895185373143 558545864083284007 and in Maple notation [-1627679928918251/1628413597910449, -11398894202880203/11398895185373143, -\ 558545864080512667/558545864083284007] and in floating point [-.9995494578, -.9999999138, -1.000000000] The cut off is at j=, 1 ------------------------- This ends this article that took, 17376.598, seconds to produce ----------------------- This took, 17376.598, seconds.