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], from i=2 to i=, 7 By Shalosh B. Ekhad --------------------------------------------- The occupants of cell, [1, 2], in a standard Young tableau of shape, [n, n, n, n, n], are all the integers from, 2, to , 6 as n goes to infinity, the distribution is [[2., .6000000000], [3., .3200000000], [4., .7200000000e-1], [5., .7680000000e-\ 2], [6., .3200000000e-3]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [2.488320000, .6661708319, 1.208486661, 3.992754338] Here is a plot 0.6HH + HH + HH 0.5 HH + HH + HH + HH 0.4 HHH + HH + HH 0.3 HH + HH + HH 0.2 HH + HH + HHH + HH 0.1 HH + HHHHHHHH + HHHHHHHH ++---+---+--+---+---+--+---+---+--+---+---+--+---+---+**********************- 0 2 3 4 5 6 --------------------------------------------- The occupants of cell, [1, 3], in a standard Young tableau of shape, [n, n, n, n, n], are all the integers from, 3, to , 11 as n goes to infinity, the distribution is [[3., .2800000000], [4., .3360000000], [5., .2329600000], [6., .1075200000], [7\ ., .3456000000e-1], [8., .7705600000e-2], [9., .1146880000e-2], [10., .\ 1032192000e-3], [11., .4300800000e-5]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [4.308886221, 1.146541553, .7689196644, 3.282663316] Here is a plot + HH + HHHH H 0.3 HHH HH +HHH H + HH 0.25 H + HH + H 0.2 H + H + H 0.15 HH + H + H 0.1 HHH + HH + HH 0.05 HH + HHHHH + HHHHH +--+---+---+--+---+---+--+---+---+--+---+--+---*****************************- 0 4 6 8 10 --------------------------------------------- The occupants of cell, [1, 4], in a standard Young tableau of shape, [n, n, n, n, n], are all the integers from, 4, to , 16 as n goes to infinity, the distribution is [[4., .1120000000], [5., .2150400000], [6., .2419200000], [7., .1996800000], [8\ ., .1282176000], [9., .6537216000e-1], [10., .2658754560e-1], [11., .8567193600\ e-2], [12., .2149171200e-2], [13., .4065853440e-3], [14., .5482414080e-4], [15. , .4723310592e-5], [16., .1968046080e-6]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [6.378602026, 1.600432683, .5668778764, 3.047037017] Here is a plot + HHH + HHH HH + HH HH 0.2 H HH + H H + H H + H H + H H 0.15 H H + H HH + H H +H H 0.1 HH + H + H + HH 0.05 HH + HH + HHHH + HHHHHH ++--+-+--+-+--+-+--+-+-+--+-+--+-+--+-+-+--+-+-*****************************- 0 4 6 8 10 12 14 16 --------------------------------------------- The occupants of cell, [1, 5], in a standard Young tableau of shape, [n, n, n, n, n], are all the integers from, 5, to , 21 as n goes to infinity, the distribution is [[5., .4032000000e-1], [6., .1075200000], [7., .1658880000], [8., .1903104000], [9., .1768345600], [10., .1376428032], [11., .9096560640e-1], [12., .5126651904\ e-1], [13., .2461125427e-1], [14., .1000751432e-1], [15., .3411804684e-2], [16. , .9600128778e-3], [17., .2177446183e-3], [18., .3838926914e-4], [19., .\ 4955427570e-5], [20., .4184583281e-6], [21., .1743576367e-7]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [8.645262507, 2.033508055, .4437289930, 2.939228811] Here is a plot + HHHH 0.18 H HHHH + H H 0.16 HH HH + H H 0.14 H H + H HH 0.12 H H + H H 0.1 H H + H HH + H H 0.08 H H + H HH 0.06 H H + H HH 0.04 HH + HHH 0.02 HHH + HHHH +++-+-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-+-++-******************************- 0 6 8 10 12 14 16 18 20 --------------------------------------------- The occupants of cell, [1, 6], in a standard Young tableau of shape, [n, n, n, n, n], are all the integers from, 6, to , 26 as n goes to infinity, the distribution is [[6., .1344000000e-1], [7., .4608000000e-1], [8., .9004800000e-1], [9., .130457\ 6000], [10., .1543495680], [11., .1560477696], [12., .1378477056], [13., .10752\ 34406], [14., .7437046989e-1], [15., .4563636453e-1], [16., .2479226369e-1], [ 17., .1187155385e-1], [18., .4976967365e-2], [19., .1809648735e-2], [20., .\ 5633640540e-3], [21., .1475065607e-3], [22., .3167206471e-4], [23., .5370215211\ e-5], [24., .6763913920e-6], [25., .5646397707e-7], [26., .2352665711e-8]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [11.07309709, 2.448263149, .3585537411, 2.882914921] Here is a plot + HHHHH + H HH 0.14 H HH + HH H + H H 0.12 H H + H H 0.1 H H + H H + H H 0.08 H HH + H H 0.06 HH H + H HH + H H 0.04 H HH + H H 0.02HH HH +H HHH + HHHHH ++---+---+--+---+---+--+---+---+--+---+---+--+-*****************************- 0 10 15 20 25 --------------------------------------------- The occupants of cell, [1, 7], in a standard Young tableau of shape, [n, n, n, n, n], are all the integers from, 7, to , 31 as n goes to infinity, the distribution is [[7., .4224000000e-2], [8., .1774080000e-1], [9., .4190208000e-1], [10., .\ 7288258560e-1], [11., .1034385408], [12., .1260590285], [13., .1356949750], [14\ ., .1310971724], [15., .1146673192], [16., .9118530096e-1], [17., .6601289887e-\ 1], [18., .4347990035e-1], [19., .2600122023e-1], [20., .1406877822e-1], [21., .6854905359e-2], [22., .2988790563e-2], [23., .1156659967e-2], [24., .\ 3931539670e-3], [25., .1157637005e-3], [26., .2898484156e-4], [27., .6012943025\ e-5], [28., .9946129561e-6], [29., .1233173259e-6], [30., .1021772129e-7], [31. , .4257383871e-9]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [13.63618427, 2.846602261, .2951844520, 2.851573397] Here is a plot + HHHH + HH HH 0.12 H HH + H H + H HH 0.1 H H + H H + H HH 0.08 H H + H H + H H 0.06 H H + H HH + H H 0.04 H HH + H H + H HH 0.02 H HH + HH HHHH +HHH HHHH ++---+--+--+--+--+--+--+--+--+--+--+--+--+--+--*****************************- 0 10 15 20 25 30 ----------------------- This took, 3648.860, seconds.