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], 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], are all the integers from, 2, to , 8 The probability distribution is 4 (-1 + n) 16 (-1 + n) (n + 1) 30 (2 + n) (n + 1) (-1 + n) [----------, ---------------------, --------------------------------, -1 + 7 n (-1 + 7 n) (-2 + 7 n) (-2 + 7 n) (-1 + 7 n) (-3 + 7 n) 32 (n + 1) (2 + n) (3 + n) (-1 + n) -------------------------------------------, (-2 + 7 n) (-4 + 7 n) (-1 + 7 n) (-3 + 7 n) 20 (-1 + n) (n + 1) (2 + n) (3 + n) (4 + n) ------------------------------------------------------, (-5 + 7 n) (-4 + 7 n) (-3 + 7 n) (-2 + 7 n) (-1 + 7 n) 48 (n + 1) (2 + n) (3 + n) (4 + n) (5 + n) (-1 + n) -------------------------------------------------------------------, 7 (-3 + 7 n) (-6 + 7 n) (-2 + 7 n) (-5 + 7 n) (-1 + 7 n) (-4 + 7 n) (5 + n) (4 + n) (3 + n) (2 + n) (n + 1) (6 + n) -------------------------------------------------------------------] 7 (-3 + 7 n) (-6 + 7 n) (-2 + 7 n) (-5 + 7 n) (-1 + 7 n) (-4 + 7 n) and in Maple notation [4*(-1+n)/(-1+7*n), 16*(-1+n)*(n+1)/(-1+7*n)/(-2+7*n), 30*(2+n)*(n+1)*(-1+n)/(-\ 2+7*n)/(-1+7*n)/(-3+7*n), 32*(n+1)*(2+n)*(3+n)*(-1+n)/(-2+7*n)/(-4+7*n)/(-1+7*n )/(-3+7*n), 20*(-1+n)*(n+1)*(2+n)*(3+n)*(4+n)/(-5+7*n)/(-4+7*n)/(-3+7*n)/(-2+7* n)/(-1+7*n), 48/7*(n+1)*(2+n)*(3+n)*(4+n)*(5+n)*(-1+n)/(-3+7*n)/(-6+7*n)/(-2+7* n)/(-5+7*n)/(-1+7*n)/(-4+7*n), 1/7*(5+n)*(4+n)*(3+n)*(2+n)*(n+1)/(-3+7*n)/(-6+7 *n)/(-2+7*n)/(-5+7*n)/(-1+7*n)/(-4+7*n)*(6+n)] `The average is` 128/7*(8*n-5)*(4*n-1)*(2*n-1)*(8*n-1)*(4*n-3)*(8*n-3)/(-3+7*n)/(-6+7*n)/(-2+7*n )/(-5+7*n)/(-1+7*n)/(-4+7*n) and the variance is 384/49*(-1+n)*(8*n-3)*(8*n-1)*(5*n-2)*(2*n-1)*(4*n-1)*(8*n-5)*(n+1)*(44691*n^4-\ 71190*n^3+37061*n^2-6930*n+360)/(-3+7*n)^2/(-6+7*n)^2/(-2+7*n)^2/(-5+7*n)^2/(-1 +7*n)^2/(-4+7*n)^2 as n goes to infinity, the distribution is [[2., .5714285714], [3., .3265306122], [4., .8746355685e-1], [5., .1332778009e-\ 1], [6., .1189980365e-2], [7., .5828475259e-4], [8., .1214265679e-5]] The limiting average, standard deviation up to the, 4, -th scaled-moment are 1/2 1/2 2097152 1536 148970 1298201530813 148970 5293996577909094089 [-------, --------------, -----------------------, -------------------] 823543 823543 409044066508800 1256583372315033600 and in floating-point [2.546499697, .7198704467, 1.224958649, 4.213008619] Here is a plot +H + HH 0.5 HH + H + HH + HH 0.4 H + HH + H + HH 0.3 H + HH + HH 0.2 H + HH + H + HH 0.1 HHH + HHHH + HHHHH ++--+-+--+-+--+-+--+-+-+--+-+--+-+--+-**************************************- 0 2 3 4 5 6 7 8 --------------------------------------------- The occupants of cell, [1, 3], in a standard Young tableau of shape, [n, n, n, n, n, n, n], are all the integers from, 3, to , 15 The probability distribution is 12 (-2 + n) (-1 + n) 108 (-1 + n) (-2 + n) (n + 1) [---------------------, --------------------------------, (-1 + 7 n) (-2 + 7 n) (-2 + 7 n) (-1 + 7 n) (-3 + 7 n) 72 (-2 + n) (-1 + n) (n + 1) (8 n + 9) -------------------------------------------, (-2 + 7 n) (-4 + 7 n) (-1 + 7 n) (-3 + 7 n) 720 (2 + n) (n + 1) (-2 + n) (-1 + n) (3 n + 2) ------------------------------------------------------, (-5 + 7 n) (-4 + 7 n) (-3 + 7 n) (-2 + 7 n) (-1 + 7 n) 2 270 (-2 + n) (-1 + n) (n + 1) (2 + n) (157 n + 413 n + 120) -------------------------------------------------------------------, 7 (-3 + 7 n) (-6 + 7 n) (-2 + 7 n) (-5 + 7 n) (-1 + 7 n) (-4 + 7 n) 2 54 (-2 + n) (3 + n) (2 + n) (n + 1) (241 n + 489 n + 60) -------------------------------------------------------------------, 8 7 (-1 + 7 n) (-2 + 7 n) (-3 + 7 n) (-4 + 7 n) (-5 + 7 n) (-6 + 7 n) 3 2 (-2 + n) (2 + n) (n + 1) (3 + n) (386 n + 1875 n + 1969 n + 60)/( (-1 + 7 n) (-2 + 7 n) (-3 + 7 n) (-4 + 7 n) (-5 + 7 n) (-6 + 7 n) (-8 + 7 n)), 432 (4 + n) (n + 1) (2 + n) (3 + n) (-2 + n) n 2 (64 n + 263 n + 219)/(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)), 180 n (-2 + n) (4 + n) 2 2 (3 + n) (2 + n) (n + 1) (149 n + 1009 n + 1572)/(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) 2 (-10 + 7 n)), 660 n (-2 + n) (29 n + 111) (5 + n) (4 + n) (3 + n) (2 + n) 2 (n + 1) /(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) (-11 + 7 n)), 2376 (-2 + n) 2 2 2 (4 n + 21) (5 + n) (4 + n) (3 + n) (2 + n) (n + 1) n/(7 (-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)), 20592 n (6 + n) (5 + n) 2 2 2 2 (4 + n) (n + 1) (2 + n) (3 + n) (-2 + n)/(49 (-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 2 (-2 + 7 n) (-1 + 7 n) (-12 + 7 n) (-13 + 7 n)), 429 (n + 1) (2 + n) 2 2 2 (3 + n) (4 + n) (5 + n) (6 + n) n/(49 (-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 + 7 n) (-13 + 7 n))] and in Maple notation [12*(-2+n)*(-1+n)/(-1+7*n)/(-2+7*n), 108*(-1+n)*(-2+n)*(n+1)/(-2+7*n)/(-1+7*n)/ (-3+7*n), 72*(-2+n)*(-1+n)*(n+1)*(8*n+9)/(-2+7*n)/(-4+7*n)/(-1+7*n)/(-3+7*n), 720*(2+n)*(n+1)*(-2+n)*(-1+n)*(3*n+2)/(-5+7*n)/(-4+7*n)/(-3+7*n)/(-2+7*n)/(-1+7 *n), 270/7*(-2+n)*(-1+n)*(n+1)*(2+n)*(157*n^2+413*n+120)/(-3+7*n)/(-6+7*n)/(-2+ 7*n)/(-5+7*n)/(-1+7*n)/(-4+7*n), 54/7*(-2+n)*(3+n)*(2+n)*(n+1)*(241*n^2+489*n+ 60)/(-1+7*n)/(-2+7*n)/(-3+7*n)/(-4+7*n)/(-5+7*n)/(-6+7*n), 8*(-2+n)*(2+n)*(n+1) *(3+n)*(386*n^3+1875*n^2+1969*n+60)/(-1+7*n)/(-2+7*n)/(-3+7*n)/(-4+7*n)/(-5+7*n )/(-6+7*n)/(-8+7*n), 432/7*(4+n)*(n+1)*(2+n)*(3+n)*(-2+n)*n*(64*n^2+263*n+219)/ (-1+7*n)/(-2+7*n)/(-3+7*n)/(-4+7*n)/(-5+7*n)/(-6+7*n)/(-8+7*n)/(-9+7*n), 180/7* n*(-2+n)*(4+n)*(3+n)*(2+n)*(n+1)^2*(149*n^2+1009*n+1572)/(-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), 660/7*n*(-2+n)*(29 *n+111)*(5+n)*(4+n)*(3+n)*(2+n)^2*(n+1)^2/(-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), 2376/7*(-2+n)*(4*n+21)* (5+n)*(4+n)*(3+n)^2*(2+n)^2*(n+1)^2*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), 20592/49*n*(6+n)* (5+n)*(4+n)^2*(n+1)^2*(2+n)^2*(3+n)^2*(-2+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+7*n)/(-13+7*n), 429/49*(n+1)^2*(2+n)^2*(3+n)^2*(4+n)^2*(5+n)^2*(6+n)*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+7*n)/( -13+7*n)] `The average is` 45/49*(67720927949*n^12-763545401308*n^11+3818685159243*n^10-11164299014166*n^9 +21169178170143*n^8-27305907463272*n^7+24445413276589*n^6-15215912934974*n^5+ 6491501223708*n^4-1836099095400*n^3+323434493568*n^2-31409665920*n+1245404160)/ (-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+7*n)/(-13+7*n) and the variance is 90/2401*(-2+n)*(n+1)*(8153933986312459419295*n^22-165050248301506377917361*n^21 +1570118635697709797574095*n^20-9329967764251956964636191*n^19+ 38824091106113708790141000*n^18-120215056480054157382371154*n^17+ 287300840155439719422366046*n^16-542522124446325179594767902*n^15+ 821977103477270468569149935*n^14-1009065596574749840691491637*n^13+ 1009397670458192632976759907*n^12-824713819519782311576846883*n^11+ 550015556250398248855054330*n^10-298381767858819704329094520*n^9+ 130823229758867816929549568*n^8-45899018026504852942862064*n^7+ 12704330158082885536601760*n^6-2719192900481293299625728*n^5+ 437429552809555231248384*n^4-50712578146338440232960*n^3+3964324524000726343680 *n^2-185291517068859801600*n+3877578804363264000)/(-11+7*n)^2/(-10+7*n)^2/(-9+7 *n)^2/(-8+7*n)^2/(-6+7*n)^2/(-5+7*n)^2/(-4+7*n)^2/(-3+7*n)^2/(-2+7*n)^2/(-1+7*n )^2/(-12+7*n)^2/(-13+7*n)^2 as n goes to infinity, the distribution is [[3., .2448979592], [4., .3148688047], [5., .2399000416], [6., .1285178795], [7\ ., .5147272213e-1], [8., .1580245355e-1], [9., .3749652416e-2], [10., .\ 6851432141e-3], [11., .9494637174e-4], [12., .9679735566e-5], [13., .6866413406\ e-6], [14., .3036169193e-7], [15., .6325352486e-9]] The limiting average, standard deviation up to the, 4, -th scaled-moment are 1/2 3047441757705 15 3261573594524983767718 [-------------, ----------------------------, 678223072849 678223072849 1/2 5539666162970350584407911294425323 3261573594524983767718 ------------------------------------------------------------, 398919836718848370915409681250677011709782150 1576627556260519926138178932575398841027376037 ----------------------------------------------] 460292119290978889517780401443088859665133250 and in floating-point [4.493273496, 1.263084164, .7930700555, 3.425276015] Here is a plot + H 0.3 HHHH + H H + HH HH 0.25H H + H + H 0.2 HH + H + H 0.15 H + H + H + H 0.1 H + HH + H 0.05 HHH + HHH + HHHHH +--+-+--+-+-+--+-+--+-+--+-+-+--+-+-****************************************+ 0 4 6 8 10 12 14 ----------------------- This took, 537.692, seconds.