The Distribution of the Occupant of Cell [1,i] in a random Standard Young ta\ bleau of shape, [n, n, n, n, n, n, n, n], and its Limiting behavior as n goes to infinity for i from 2 to, 3 By Shalosh B. Ekhad --------------------------------------------- The occupants of cell, [1, 2], in a standard Young tableau of shape, [n, n, n, n, n, n, n, n], are all the integers from, 2, to , 9 The probability distribution is 9 (-1 + n) 21 (-1 + n) (1 + n) 189 (-1 + n) (2 + n) (1 + n) [------------, -----------------------, ----------------------------------, 2 (-1 + 8 n) 2 (-1 + 8 n) (-1 + 4 n) 8 (-3 + 8 n) (-1 + 8 n) (-1 + 4 n) 63 (2 + n) (3 + n) (1 + n) (-1 + n) ---------------------------------------------, 8 (-3 + 8 n) (-1 + 8 n) (-1 + 2 n) (-1 + 4 n) 105 (-1 + n) (1 + n) (2 + n) (3 + n) (4 + n) ---------------------------------------------------------, 16 (-5 + 8 n) (-1 + 2 n) (-3 + 8 n) (-1 + 4 n) (-1 + 8 n) 27 (-1 + n) (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) --------------------------------------------------------------------, 63 16 (-3 + 4 n) (-5 + 8 n) (-1 + 2 n) (-3 + 8 n) (-1 + 4 n) (-1 + 8 n) (-1 + n) (3 + n) (4 + n) (5 + n) (6 + n) (2 + n) (1 + n)/(128 (-7 + 8 n) (-3 + 4 n) (-5 + 8 n) (-1 + 2 n) (-3 + 8 n) (-1 + 4 n) (-1 + 8 n)), (4 + n) (3 + n) (2 + n) (1 + n) (5 + n) (6 + n) (7 + n)/(128 (-1 + 8 n) (-1 + 4 n) (-3 + 8 n) (-1 + 2 n) (-5 + 8 n) (-3 + 4 n) (-7 + 8 n))] and in Maple notation [9/2*(-1+n)/(-1+8*n), 21/2*(-1+n)*(1+n)/(-1+8*n)/(-1+4*n), 189/8*(-1+n)*(2+n)*( 1+n)/(-3+8*n)/(-1+8*n)/(-1+4*n), 63/8*(2+n)*(3+n)*(1+n)*(-1+n)/(-3+8*n)/(-1+8*n )/(-1+2*n)/(-1+4*n), 105/16*(-1+n)*(1+n)*(2+n)*(3+n)*(4+n)/(-5+8*n)/(-1+2*n)/(-\ 3+8*n)/(-1+4*n)/(-1+8*n), 27/16*(-1+n)*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)/(-3+4*n)/( -5+8*n)/(-1+2*n)/(-3+8*n)/(-1+4*n)/(-1+8*n), 63/128*(-1+n)*(3+n)*(4+n)*(5+n)*(6 +n)*(2+n)*(1+n)/(-7+8*n)/(-3+4*n)/(-5+8*n)/(-1+2*n)/(-3+8*n)/(-1+4*n)/(-1+8*n), 1/128*(4+n)*(3+n)*(2+n)*(1+n)/(-1+8*n)/(-1+4*n)/(-3+8*n)/(-1+2*n)/(-5+8*n)/(-3+ 4*n)/(-7+8*n)*(5+n)*(6+n)*(7+n)] `The average is` 81/128*(9*n-7)*(9*n-5)*(3*n-1)*(9*n-1)*(3*n-2)*(9*n-4)*(9*n-2)/(-1+8*n)/(-1+4*n )/(-3+8*n)/(-1+2*n)/(-5+8*n)/(-3+4*n)/(-7+8*n) and the variance is 81/16384*(-1+n)*(9*n-1)*(9*n-2)*(3*n-1)*(9*n-4)*(9*n-5)*(3*n-2)*(1+n)*(32009911 *n^6-78811677*n^5+75732961*n^4-35823879*n^3+8600560*n^2-955332*n+35280)/(-1+8*n )^2/(-1+4*n)^2/(-3+8*n)^2/(-1+2*n)^2/(-5+8*n)^2/(-3+4*n)^2/(-7+8*n)^2 as n goes to infinity, the distribution is [[2., .5625000000], [3., .3281250000], [4., .9228515625e-1], [5., .1538085938e-\ 1], [6., .1602172852e-2], [7., .1029968262e-3], [8., .3755092621e-5], [9., .\ 5960464478e-7]] The limiting average, standard deviation up to the, 4, -th scaled-moment are 1/2 1/2 43046721 2187 32009911 1461979241852726 32009911 [--------, ----------------, ----------------------------, 16777216 16777216 6722626313017389681 566147374363988846112103 ------------------------] 132321453719121281091123 and in floating-point [2.565784514, .7375154461, 1.230394748, 4.278575835] Here is a plot +H + HH 0.5 H + HH + H + H 0.4 HH + H + HH 0.3 H + H + HH + H 0.2 H + H + H 0.1 H + HHHH + HHH + HHHHH ++--+-+-+-+-+-+-+-+-+-+-+--+-+-+-+******************************************- 0 2 3 4 5 6 7 8 9 --------------------------------------------- The occupants of cell, [1, 3], in a standard Young tableau of shape, [n, n, n, n, n, n, n, n], are all the integers from, 3, to , 17 The probability distribution is 15 (-1 + n) (-2 + n) 315 (-1 + n) (-2 + n) (1 + n) [-----------------------, ----------------------------------, 2 (-1 + 8 n) (-1 + 4 n) 4 (-1 + 8 n) (-1 + 4 n) (-3 + 8 n) 21 (-1 + n) (-2 + n) (47 n + 54) (1 + n) ---------------------------------------------, 8 (-1 + 8 n) (-1 + 4 n) (-3 + 8 n) (-1 + 2 n) 315 (7 n + 5) (-1 + n) (-2 + n) (2 + n) (1 + n) --------------------------------------------------------, 4 (-1 + 8 n) (-1 + 4 n) (-3 + 8 n) (-1 + 2 n) (-5 + 8 n) 2 135 (-1 + n) (-2 + n) (2 + n) (1 + n) (111 n + 301 n + 100) --------------------------------------------------------------------, 105 16 (-1 + 8 n) (-1 + 4 n) (-3 + 8 n) (-1 + 2 n) (-5 + 8 n) (-3 + 4 n) 2 (-1 + n) (-2 + n) (3 + n) (2 + n) (1 + n) (1519 n + 3271 n + 540)/(64 (-1 + 8 n) (-1 + 4 n) (-3 + 8 n) (-1 + 2 n) (-5 + 8 n) (-3 + 4 n) (-7 + 8 n)), 35 (-2 + n) (3 + n) (2 + n) (1 + n) 3 2 (2407 n + 12129 n + 13742 n + 840)/(128 (-1 + 8 n) (-1 + 4 n) (-3 + 8 n) (-1 + 2 n) (-5 + 8 n) (-3 + 4 n) (-7 + 8 n)), 9 (-2 + n) (4 + n) (3 + n) 3 2 (2 + n) (1 + n) (3943 n + 17214 n + 15761 n + 210)/(32 (-9 + 8 n) (-1 + 8 n) (-1 + 4 n) (-3 + 8 n) (-1 + 2 n) (-5 + 8 n) (-3 + 4 n) (-7 + 8 n)), 105 (-2 + n) (4 + n) (3 + n) (2 + n) (1 + n) 3 2 (1811 n + 14718 n + 34897 n + 22830) n/(256 (-9 + 8 n) (-1 + 8 n) (-5 + 4 n) (-1 + 4 n) (-3 + 8 n) (-1 + 2 n) (-5 + 8 n) (-3 + 4 n) (-7 + 8 n)), 1155 n (-2 + n) (5 + n) (4 + n) (3 + n) (2 + n) 2 2 (347 n + 2191 n + 3174) (1 + n) /(512 (-9 + 8 n) (-1 + 8 n) (-5 + 4 n) (-1 + 4 n) (-11 + 8 n) (-3 + 8 n) (-1 + 2 n) (-5 + 8 n) (-3 + 4 n) 2 (-7 + 8 n)), 3465 (-2 + n) (5 + n) (4 + n) (3 + n) (47 n + 419 n + 894) 2 2 (2 + n) (1 + n) n/(1024 (-9 + 8 n) (-1 + 8 n) (-5 + 4 n) (-1 + 4 n) (-11 + 8 n) (-3 + 8 n) (-3 + 2 n) (-1 + 2 n) (-5 + 8 n) (-3 + 4 n) 2 (-7 + 8 n)), 2145 n (-2 + n) (6 + n) (5 + n) (4 + n) (23 n + 113) (3 + n) 2 2 (2 + n) (1 + n) /(512 (-9 + 8 n) (-1 + 8 n) (-5 + 4 n) (-1 + 4 n) (-11 + 8 n) (-3 + 8 n) (-3 + 2 n) (-1 + 2 n) (-13 + 8 n) (-5 + 8 n) 2 (-3 + 4 n) (-7 + 8 n)), 3003 (5 + n) (-2 + n) (7 n + 44) (6 + n) (4 + n) 2 2 2 (3 + n) (2 + n) (1 + n) n/(1024 (-9 + 8 n) (-1 + 8 n) (-5 + 4 n) (-1 + 4 n) (-11 + 8 n) (-3 + 8 n) (-3 + 2 n) (-1 + 2 n) (-13 + 8 n) (-5 + 8 n) (-7 + 4 n) (-3 + 4 n) (-7 + 8 n)), 45045 n (-2 + n) (7 + n) 2 2 2 2 2 (6 + n) (5 + n) (4 + n) (3 + n) (2 + n) (1 + n) /(8192 (-7 + 8 n) (-15 + 8 n) (-3 + 4 n) (-7 + 4 n) (-5 + 8 n) (-13 + 8 n) (-1 + 2 n) (-3 + 2 n) (-3 + 8 n) (-11 + 8 n) (-1 + 4 n) (-5 + 4 n) (-1 + 8 n) 2 2 2 2 2 (-9 + 8 n)), 715 n (7 + n) (6 + n) (5 + n) (4 + n) (3 + n) (2 + n) 2 (1 + n) /(8192 (-7 + 8 n) (-15 + 8 n) (-3 + 4 n) (-7 + 4 n) (-5 + 8 n) (-13 + 8 n) (-1 + 2 n) (-3 + 2 n) (-3 + 8 n) (-11 + 8 n) (-1 + 4 n) (-5 + 4 n) (-1 + 8 n) (-9 + 8 n))] and in Maple notation [15/2*(-1+n)*(-2+n)/(-1+8*n)/(-1+4*n), 315/4*(-1+n)*(-2+n)*(1+n)/(-1+8*n)/(-1+4 *n)/(-3+8*n), 21/8*(-1+n)*(-2+n)*(47*n+54)*(1+n)/(-1+8*n)/(-1+4*n)/(-3+8*n)/(-1 +2*n), 315/4*(7*n+5)*(-1+n)*(-2+n)*(2+n)*(1+n)/(-1+8*n)/(-1+4*n)/(-3+8*n)/(-1+2 *n)/(-5+8*n), 135/16*(-1+n)*(-2+n)*(2+n)*(1+n)*(111*n^2+301*n+100)/(-1+8*n)/(-1 +4*n)/(-3+8*n)/(-1+2*n)/(-5+8*n)/(-3+4*n), 105/64*(-1+n)*(-2+n)*(3+n)*(2+n)*(1+ n)*(1519*n^2+3271*n+540)/(-1+8*n)/(-1+4*n)/(-3+8*n)/(-1+2*n)/(-5+8*n)/(-3+4*n)/ (-7+8*n), 35/128*(-2+n)*(3+n)*(2+n)*(1+n)*(2407*n^3+12129*n^2+13742*n+840)/(-1+ 8*n)/(-1+4*n)/(-3+8*n)/(-1+2*n)/(-5+8*n)/(-3+4*n)/(-7+8*n), 9/32*(-2+n)*(4+n)*( 3+n)*(2+n)*(1+n)*(3943*n^3+17214*n^2+15761*n+210)/(-9+8*n)/(-1+8*n)/(-1+4*n)/(-\ 3+8*n)/(-1+2*n)/(-5+8*n)/(-3+4*n)/(-7+8*n), 105/256*(-2+n)*(4+n)*(3+n)*(2+n)*(1 +n)*(1811*n^3+14718*n^2+34897*n+22830)*n/(-9+8*n)/(-1+8*n)/(-5+4*n)/(-1+4*n)/(-\ 3+8*n)/(-1+2*n)/(-5+8*n)/(-3+4*n)/(-7+8*n), 1155/512*n*(-2+n)*(5+n)*(4+n)*(3+n) *(2+n)*(347*n^2+2191*n+3174)*(1+n)^2/(-9+8*n)/(-1+8*n)/(-5+4*n)/(-1+4*n)/(-11+8 *n)/(-3+8*n)/(-1+2*n)/(-5+8*n)/(-3+4*n)/(-7+8*n), 3465/1024*(-2+n)*(5+n)*(4+n)* (3+n)*(47*n^2+419*n+894)*(2+n)^2*(1+n)^2*n/(-9+8*n)/(-1+8*n)/(-5+4*n)/(-1+4*n)/ (-11+8*n)/(-3+8*n)/(-3+2*n)/(-1+2*n)/(-5+8*n)/(-3+4*n)/(-7+8*n), 2145/512*n*(-2 +n)*(6+n)*(5+n)*(4+n)*(23*n+113)*(3+n)^2*(2+n)^2*(1+n)^2/(-9+8*n)/(-1+8*n)/(-5+ 4*n)/(-1+4*n)/(-11+8*n)/(-3+8*n)/(-3+2*n)/(-1+2*n)/(-13+8*n)/(-5+8*n)/(-3+4*n)/ (-7+8*n), 3003/1024*(5+n)*(-2+n)*(7*n+44)*(6+n)*(4+n)^2*(3+n)^2*(2+n)^2*(1+n)^2 *n/(-9+8*n)/(-1+8*n)/(-5+4*n)/(-1+4*n)/(-11+8*n)/(-3+8*n)/(-3+2*n)/(-1+2*n)/(-\ 13+8*n)/(-5+8*n)/(-7+4*n)/(-3+4*n)/(-7+8*n), 45045/8192*n*(-2+n)*(7+n)*(6+n)*(5 +n)^2*(4+n)^2*(3+n)^2*(2+n)^2*(1+n)^2/(-7+8*n)/(-15+8*n)/(-3+4*n)/(-7+4*n)/(-5+ 8*n)/(-13+8*n)/(-1+2*n)/(-3+2*n)/(-3+8*n)/(-11+8*n)/(-1+4*n)/(-5+4*n)/(-1+8*n)/ (-9+8*n), 715/8192*n*(7+n)*(6+n)^2*(5+n)^2*(4+n)^2*(3+n)^2*(2+n)^2*(1+n)^2/(-7+ 8*n)/(-15+8*n)/(-3+4*n)/(-7+4*n)/(-5+8*n)/(-13+8*n)/(-1+2*n)/(-3+2*n)/(-3+8*n)/ (-11+8*n)/(-1+4*n)/(-5+4*n)/(-1+8*n)/(-9+8*n)] `The average is` 5/8192*(128245678505551*n^14-1698683404378793*n^13+10164179229003275*n^12-\ 36330150647512141*n^11+86433274007117205*n^10-144361796173842387*n^9+ 173977474812274697*n^8-153106672557243391*n^7+98442173674291384*n^6-\ 45786397337368520*n^5+15072602438165328*n^4-3379989803219568*n^3+ 483371716354560*n^2-38940932217600*n+1307674368000)/(-7+8*n)/(-15+8*n)/(-3+4*n) /(-7+4*n)/(-5+8*n)/(-13+8*n)/(-1+2*n)/(-3+2*n)/(-3+8*n)/(-11+8*n)/(-1+4*n)/(-5+ 4*n)/(-1+8*n)/(-9+8*n) and the variance is 5/67108864*(-2+n)*(1+n)*(6722443357345829868734951963*n^26-\ 162137515027097676013992610911*n^25+1855641812299836540247628397584*n^24-\ 13407053579139683251370567760498*n^23+68629376559349427362251281824049*n^22-\ 264801711964277743798611065806809*n^21+799949015599073482374292730018306*n^20-\ 1940099933642987025393011659266696*n^19+3843150527762487717603854846777381*n^18 -6293184616359810437759975940811857*n^17+8589674130208091079897163550872268*n^ 16-9826079311064538059158952402079834*n^15+9450621155080889138028917226406823*n ^14-7651366323618207088813111888659975*n^13+5211366478366484369784753559547810* n^12-2978953180671757244710103178685452*n^11+1423105754648126277651042402341000 *n^10-564601605974320407978686697831888*n^9+184415801930748262726048251515232*n ^8-49013579636434684032522021587520*n^7+10434818418886894229109771398784*n^6-\ 1742149007305132754436207298560*n^5+221466293964349390199294668800*n^4-\ 20535770784866899568670720000*n^3+1298409732224177860546560000*n^2-\ 49583906481595156070400000*n+855006126362099712000000)/(-7+8*n)^2/(-15+8*n)^2/( -3+4*n)^2/(-7+4*n)^2/(-5+8*n)^2/(-13+8*n)^2/(-1+2*n)^2/(-3+2*n)^2/(-3+8*n)^2/(-\ 11+8*n)^2/(-1+4*n)^2/(-5+4*n)^2/(-1+8*n)^2/(-9+8*n)^2 as n goes to infinity, the distribution is [[3., .2343750000], [4., .3076171875], [5., .2409667969], [6., .1345825195], [7\ ., .5716323853e-1], [8., .1901328564e-1], [9., .5021393299e-2], [10., .\ 1057595015e-2], [11., .1770956442e-3], [12., .2332875738e-4], [13., .2369852155\ e-5], [14., .1794796844e-6], [15., .9559244063e-8], [16., .3200639753e-9], [17. , .5080380561e-11]] The limiting average, standard deviation up to the, 4, -th scaled-moment are 1/2 641228392527755 33612216786729149343674759815 [---------------, --------------------------------, 140737488355328 140737488355328 1/2 197425241597859653081076001590439346143062 33612216786729149343674759815 /45191244692723072856578517932876153687241416155917553369, 783758298147235432287629900582059088519984539794827157833 ---------------------------------------------------------] 225956223463615364282892589664380768436207080779587766845 and in floating-point [4.556201763, 1.302683108, .8009344112, 3.468628950] Here is a plot 0.3 H + HH H + H H + HH H 0.25H HH +H H + H 0.2 H + H + H 0.15 H + H + H + H 0.1 H + H + H 0.05 HHH + HHH + HHHH +--+-+-+-+-+-+-+-+-+-+-+-+--+-+-********************************************+ 0 4 6 8 10 12 14 16 ----------------------- This took, 6273.054, seconds.