The Limiting Distribution, as n goes to infinity, of the Occupant of Cell [1\ ,i] in a random Standard Young tableau of shape, [n, n, n, n, n, n, n, n], from i=2 to i=, 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 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 [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 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 [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, 6343.908, seconds.