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], from i=2 to i=, 10 By Shalosh B. Ekhad --------------------------------------------- The occupants of cell, [1, 2], in a standard Young tableau of shape, [n, n, n, n], are all the integers from, 2, to , 5 as n goes to infinity, the distribution is [[2., .6250000000], [3., .3125000000], [4., .5859375000e-1], [5., .3906250000e-\ 2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [2.441406250, .6222473562, 1.195326067, 3.784229977] Here is a plot +HH 0.6 HH + HHH + HHH 0.5 HH + HHH + H 0.4 HHH + HH + HHH 0.3 HHH + HHH + HHH + HHH 0.2 HH + HHH + HHH 0.1 HHH + HHHHHH + HHHHHHHHHHHHH ++--+-+--+-+--+-+--+-+-+--+-+--+-+--+-+-+--+-+--+-+--+-+-+--+-+--+-*********- 0 2 2.5 3 3.5 4 4.5 5 --------------------------------------------- The occupants of cell, [1, 3], in a standard Young tableau of shape, [n, n, n, n], are all the integers from, 3, to , 9 as n goes to infinity, the distribution is [[3., .3125000000], [4., .3515625000], [5., .2226562500], [6., .8789062500e-1], [7., .2197265625e-1], [8., .3204345703e-2], [9., .2136230469e-3]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [4.165740967, 1.055278492, .7488756366, 3.150686786] Here is a plot 0.35 HHHH + HHHHH HH +HHHHH HH 0.3 HH + HH + H 0.25 HH + HH + H 0.2 HH + H 0.15 HH + H + HH 0.1 HH + HH + HHHH 0.05 HH + HHH + HHHHHHHHHH ++--+-+--+-+--+-+--+-+-+--+-+--+-+--+-+-+--+-+--+-+--+-+-+-*****************- 0 3 4 5 6 7 8 9 --------------------------------------------- The occupants of cell, [1, 4], in a standard Young tableau of shape, [n, n, n, n], are all the integers from, 4, to , 13 as n goes to infinity, the distribution is [[4., .1367187500], [5., .2460937500], [6., .2563476562], [7., .1916503906], [8\ ., .1070251465], [9., .4486083984e-1], [10., .1385879517e-1], [11., .3004074097\ e-2], [12., .4130601883e-3], [13., .2753734589e-4]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [6.093878627, 1.456267887, .5447252587, 2.947072093] Here is a plot 0.25 HHHHHHH + HH H + HH HH + H HH 0.2 H HH + H H + H H + H HH 0.15HH H +H HH + H + H 0.1 HH + HH + H + HH 0.05 H + HHHH + HHH + HHHHHHH ++---+--+--+--+---+--+--+--+---+--+--+--+---+--+--+--+--********************- 0 4 6 8 10 12 --------------------------------------------- The occupants of cell, [1, 5], in a standard Young tableau of shape, [n, n, n, n], are all the integers from, 5, to , 17 as n goes to infinity, the distribution is [[5., .5468750000e-1], [6., .1367187500], [7., .1962890625], [8., .2072143555], [9., .1741027832], [10., .1192016602], [11., .6671905518e-1], [12., .3030681610\ e-1], [13., .1097559929e-1], [14., .3068447113e-2], [15., .6264746189e-3], [16. , .8390285075e-4], [17., .5593523383e-5]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [8.178497037, 1.833345178, .4208365932, 2.857611640] Here is a plot + HHHHH 0.2 HHH HH + H HH + HH HH + H H + H H 0.15 H H + H H + H HH + H H 0.1 H H + H H + H H + H H + H HH 0.05 HH + HH + HHH + HHH + HHHHHH +--+-+--+-+-+--+-+--+-+--+-+-+--+-+--+-+--+-+--+-+-+--+*********************+ 0 6 8 10 12 14 16 --------------------------------------------- The occupants of cell, [1, 6], in a standard Young tableau of shape, [n, n, n, n], are all the integers from, 6, to , 21 as n goes to infinity, the distribution is [[6., .2050781250e-1], [7., .6591796875e-1], [8., .1201629639], [9., .161285400\ 4], [10., .1750946045], [11., .1601943970], [12., .1255211234], [13., .\ 8452981710e-1], [14., .4875954241e-1], [15., .2388434485e-1], [16., .9791462682\ e-2], [17., .3284197301e-2], [18., .8702423656e-3], [19., .1719259308e-3], [20. , .2268467142e-4], [21., .1512311428e-5]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [10.38838042, 2.190498679, .3354427968, 2.813330240] Here is a plot + HHHH + HHH HHH 0.16 HH HH + H H 0.14 H HH + HH H 0.12 H H + H H + H H 0.1 H H + H HH 0.08 H H + HH H 0.06 H H + H HH 0.04 H HH + H HH + H HH 0.02 HHHH + HHHHH ++-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-++-+-+-+-+-+-+-+-**********************- 0 6 8 10 12 14 16 18 20 --------------------------------------------- The occupants of cell, [1, 7], in a standard Young tableau of shape, [n, n, n, n], are all the integers from, 7, to , 25 as n goes to infinity, the distribution is [[7., .7324218750e-2], [8., .2883911133e-1], [9., .6362915039e-1], [10., .10290\ 52734], [11., .1349687576], [12., .1507669687], [13., .1471117139], [14., .1268\ 930733], [15., .9712594002e-1], [16., .6588870892e-1], [17., .3942834679e-1], [ 18., .2064809203e-1], [19., .9352982888e-2], [20., .3603838133e-2], [21., .\ 1153137464e-2], [22., .2955308582e-3], [23., .5718427587e-4], [24., .7472945143\ e-5], [25., .4981963428e-6]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [12.70157682, 2.530539238, .2720781852, 2.790517743] Here is a plot + HHHHHH 0.14 HH H + HH HH + H HH 0.12 H H + HH H + H HH 0.1 H H + H H 0.08 HH H + H HH + H H 0.06 HH H + H H + H H 0.04 H HH + H H 0.02 HH HH + H HHH +HH HHHHH +++-+-++-++-+-++-+-++-++-+-++-++-+-++-+-++-++-+-++-+-+**********************- 0 8 10 12 14 16 18 20 22 24 --------------------------------------------- The occupants of cell, [1, 8], in a standard Young tableau of shape, [n, n, n, n], are all the integers from, 8, to , 29 as n goes to infinity, the distribution is [[8., .2517700195e-2], [9., .1174926758e-1], [10., .3040122986e-1], [11., .\ 5732059479e-1], [12., .8749336004e-1], [13., .1140144467], [14., .1306977347], [15., .1340927370], [16., .1242810977], [17., .1044590492], [18., .7963071403e-\ 1], [19., .5491568934e-1], [20., .3409858987e-1], [21., .1893245873e-1], [22., .9311269956e-2], [23., .4004920978e-2], [24., .1480390433e-2], [25., .\ 4588388318e-3], [26., .1149484270e-3], [27., .2192582863e-4], [28., .2846141216\ e-5], [29., .1897427478e-6]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [15.10192441, 2.855649373, .2227216083, 2.779051022] Here is a plot + HHHHHH + H HH 0.12 HH H + H H + H HH 0.1 H H + H H + H HH 0.08 H H + H H + H H 0.06 HH H + H H + H H 0.04 H H + HH HH + H H 0.02 H HH + HH HH + HHH HHHHH +*---+--+---+--+---+--+---+--+---+--+---+--+---+--+---+*********************- 0 10 15 20 25 --------------------------------------------- The occupants of cell, [1, 9], in a standard Young tableau of shape, [n, n, n, n], are all the integers from, 9, to , 33 as n goes to infinity, the distribution is [[9., .8392333984e-3], [10., .4531860352e-2], [11., .1344680786e-1], [12., .\ 2890110016e-1], [13., .5012190342e-1], [14., .7415413856e-1], [15., .9668998420\ e-1], [16., .1133805374], [17., .1210857066], [18., .1186573729], [19., .107094\ 2252], [20., .8910722245e-1], [21., .6826864028e-1], [22., .4802276198e-1], [23\ ., .3088152539e-1], [24., .1804738897e-1], [25., .9511360770e-2], [26., .\ 4475275102e-2], [27., .1855262423e-2], [28., .6656865565e-3], [29., .2016447929\ e-3], [30., .4968960079e-4], [31., .9379329009e-5], [32., .1211496664e-5], [33. , .8076644424e-7]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [17.57710298, 3.167586343, .1829244646, 2.774001956] Here is a plot 0.12 HHHHH + HH H + H HH 0.1 HH H + H H + H HH + HH H 0.08 H HH + H H + HH H 0.06 H HH + H H + H H 0.04 HH H + H H + H HH + H H 0.02 HH HHH + HH HH + HHH HHHH +***-+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+-*********************- 0 10 15 20 25 30 --------------------------------------------- The occupants of cell, [1, 10], in a standard Young tableau of shape, [n, n, n, n], are all the integers from, 10, to , 37 as n goes to infinity, the distribution is [[10., .2727508545e-3], [11., .1673698425e-2], [12., .5599915981e-2], [13., .\ 1349657774e-1], [14., .2615008503e-1], [15., .4313623533e-1], [16., .6268983707\ e-1], [17., .8204812556e-1], [18., .9812571196e-1], [19., .1082794722], [20., .\ 1109286619], [21., .1058765794], [22., .9428085321e-1], [23., .7830575289e-1], [24., .6056165325e-1], [25., .4349293050e-1], [26., .2889126413e-1], [27., .\ 1766248019e-1], [28., .9874142028e-2], [29., .5007256229e-2], [30., .2279671496\ e-2], [31., .9193684613e-3], [32., .3225885207e-3], [33., .9603864461e-4], [34. , .2337178980e-4], [35., .4376531697e-5], [36., .5631566522e-6], [37., .\ 3754377682e-7]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [20.11745956, 3.467793624, .1499914633, 2.772748664] Here is a plot + HHHH + HH HHH 0.1 HH H + H H + HH H + H H 0.08 H H + H HH + H H 0.06 H H + HH H + H H + H H 0.04 H H + H H + HH HH 0.02 HH H + H HH + HHH HHH + HHH HHH +*****-+--+-+--+--+-+--+--+--+-+--+--+-+--+--+-+--+--+-*********************- 0 10 15 20 25 30 35 ----------------------- This took, 2837.431, seconds.