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], from i=2 to i=, 5 By Shalosh B. Ekhad --------------------------------------------- The occupants of cell, [1, 2], in a standard Young tableau of shape, [n, n, n, n, n, n], are all the integers from, 2, to , 7 as n goes to infinity, the distribution is [[2., .5833333333], [3., .3240740741], [4., .8101851852e-1], [5., .1080246914e-\ 1], [6., .7501714678e-3], [7., .2143347051e-4]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [2.521626372, .6970042174, 1.217920121, 4.123292640] Here is a plot +H + HH + HH 0.5 HH + HH + H 0.4 HH + HH + HH + HH 0.3 HH + H + HH 0.2 HH + HH + HH + HH 0.1 HHH + HHHHH + HHHHHH ++--+--+--+--+--+--+--+--+--+--+--+--+-+--+-********************************- 0 2 3 4 5 6 7 --------------------------------------------- The occupants of cell, [1, 3], in a standard Young tableau of shape, [n, n, n, n, n, n], are all the integers from, 3, to , 13 as n goes to infinity, the distribution is [[3., .2592592593], [4., .3240740741], [5., .2376543210], [6., .1200274348], [7\ ., .4415294925e-1], [8., .1202774920e-1], [9., .2417219174e-2], [10., .\ 3500800183e-3], [11., .3473016054e-4], [12., .2122398700e-5], [13., .6063996286\ e-7]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [4.413466988, 1.212750915, .7828296655, 3.366685610] Here is a plot + HH 0.3 HH HH + HH H + HH HH +HH H 0.25 HH + H + H 0.2 H + HH + H 0.15 H + H + HH 0.1 HH + H + HH 0.05 HH + HHH + HHHHHH +--+--+--+--+--+--+-+--+--+--+--+--+--+--***********************************+ 0 4 6 8 10 12 --------------------------------------------- The occupants of cell, [1, 4], in a standard Young tableau of shape, [n, n, n, n, n, n], are all the integers from, 4, to , 19 as n goes to infinity, the distribution is [[4., .9722222222e-1], [5., .1944444444], [6., .2295524691], [7., .2012602881], [8., .1400945216], [9., .7953484606e-1], [10., .3726685147e-1], [11., .\ 1446163885e-1], [12., .4636228360e-2], [13., .1218256854e-2], [14., .2587166115\ e-3], [15., .4341779229e-4], [16., .5556557708e-5], [17., .5109478352e-6], [18. , .3016011527e-7], [19., .8617175792e-9]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [6.591407801, 1.707667477, .5821536194, 3.111051319] Here is a plot + HHH + HH HH + H HHH 0.2 HH H + H H + H HH + HH H 0.15 H H + H H + H H + H H 0.1 H H +H H + H + H + H 0.05 HH + HH + HHH + HHHH ++-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-++-************************************- 0 4 6 8 10 12 14 16 18 --------------------------------------------- The occupants of cell, [1, 5], in a standard Young tableau of shape, [n, n, n, n, n, n], are all the integers from, 5, to , 25 as n goes to infinity, the distribution is [[5., .3240740741e-1], [6., .9002057613e-1], [7., .1453189300], [8., .175540123\ 5], [9., .1733451773], [10., .1453498895], [11., .1053415384], [12., .\ 6655842323e-1], [13., .3679974046e-1], [14., .1781437875e-1], [15., .7536369863\ e-2], [16., .2774819311e-2], [17., .8834642895e-3], [18., .2410051725e-3], [19. , .5561525256e-4], [20., .1066495187e-4], [21., .1656844869e-5], [22., .\ 2007416314e-6], [23., .1783387474e-7], [24., .1035805856e-8], [25., .2959445304\ e-10]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [9.000684690, 2.185379388, .4594584641, 2.991562877] Here is a plot + HHHHH + H HH 0.16 H H + HH HH 0.14 H H + H H 0.12 H H + HH H + H H 0.1 H H + H H 0.08 H H + H HH 0.06 H H + H HH 0.04H H +H HH + H 0.02 HHH + HHH ++---+---+--+---+---+--+---+---+--+---+-************************************- 0 5 10 15 20 25 ----------------------- This took, 1522.635, seconds.