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], from i=2 to i=40 By Shalosh B. Ekhad --------------------------------------------- The occupants of cell, [1, 2], in a standard Young tableau of shape, [n, n], are all the integers from, 2, to , 3 as n goes to infinity, the distribution is [[2., .7500000000], [3., .2500000000]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [2.250000000, .4330127020, 1.154700539, 2.333333333] Here is a plot +HHH + HHH 0.7 HHHH + HHHH + HHH + HHHH 0.6 HHHH + HHHH + HHH + HHHH 0.5 HHH + HHHH + HHH + HHHH 0.4 HHHH + HHHH + HHH + HHHH 0.3 HHHH + HHH ++--+--+--+--+--+--+--+--+--+--+--+--+-+--+--+--+--+--+--+--+--+--+--+--+***- 2 2.2 2.4 2.6 2.8 3 --------------------------------------------- The occupants of cell, [1, 3], in a standard Young tableau of shape, [n, n], are all the integers from, 3, to , 5 as n goes to infinity, the distribution is [[3., .5000000000], [4., .3750000000], [5., .1250000000]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [3.625000000, .6959705454, .6604840139, 2.261186264] Here is a plot 0.5HHHH + HHHHH + HHHHHH 0.45 HHHHHH + HHHH 0.4 HHHHHH + HHHHHH + HHH 0.35 HHH + HHH + HH 0.3 HH + HHH 0.25 HHH + HHH + HHH 0.2 HH + HHH + HHH 0.15 HHH ++---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+--**- 3 3.5 4 4.5 5 --------------------------------------------- The occupants of cell, [1, 4], in a standard Young tableau of shape, [n, n], are all the integers from, 4, to , 7 as n goes to infinity, the distribution is [[4., .3125000000], [5., .3750000000], [6., .2343750000], [7., .7812500000e-1]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [5.078125000, .9237269534, .4395040744, 2.276334678] Here is a plot + HHHHH + HHHHHH HHH 0.35 HHHHH HH + HHHHHH HHH +HHHHH HH 0.3 HHH + HHH + HH 0.25 HHH + HHH + HH + HH 0.2 HHH + HH + HH 0.15 HH + HHH + HH + HHH 0.1 HH ++--+-+--+-+--+-+--+-+-+--+-+--+-+--+-+-+--+-+--+-+--+-+-+--+-+--+-+--+-+-**- 4 4.5 5 5.5 6 6.5 7 --------------------------------------------- The occupants of cell, [1, 5], in a standard Young tableau of shape, [n, n], are all the integers from, 5, to , 9 as n goes to infinity, the distribution is [[5., .1875000000], [6., .3125000000], [7., .2812500000], [8., .1640625000], [9\ ., .5468750000e-1]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [6.585937500, 1.128656168, .3077237475, 2.307552426] Here is a plot + HHHHHH 0.3 HH HHHHHHH + HH HHHHHHH + HH HH + H HH 0.25 HH HH + HH HH + HH HH + HH HH 0.2 HH HH +H HH + HH + HHH 0.15 HH + HH + HH + HH 0.1 HHH + HH + HH 0.05---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+--**- 5 6 7 8 9 --------------------------------------------- The occupants of cell, [1, 6], in a standard Young tableau of shape, [n, n], are all the integers from, 6, to , 11 as n goes to infinity, the distribution is [[6., .1093750000], [7., .2343750000], [8., .2734375000], [9., .2187500000], [ 10., .1230468750], [11., .4101562500e-1]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [8.134765625, 1.316735111, .2180333283, 2.340811663] Here is a plot + HHHH + HHHHH HHH 0.25 HHHH HHH + HHHH HHH + H HHH + HH HH 0.2 H HH + HH HH + H HH + HH HH + H HH 0.15 HH HH + H HH + HH HH +H HH 0.1 H + HH + HHH + HH 0.05 HH ++--+--+--+--+--+--+--+--+--+--+--+--+-+--+--+--+--+--+--+--+--+--+--+--+-**- 6 7 8 9 10 11 --------------------------------------------- The occupants of cell, [1, 7], in a standard Young tableau of shape, [n, n], are all the integers from, 7, to , 13 as n goes to infinity, the distribution is [[7., .6250000000e-1], [8., .1640625000], [9., .2343750000], [10., .2343750000] , [11., .1757812500], [12., .9667968750e-1], [13., .3222656250e-1]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [9.715820312, 1.491611869, .1520868356, 2.372430693] Here is a plot + HHHHHHHHHHHHHHH + H HH + HH HH 0.2 HH HH + HH HHH + HH HH + HH HH + H H 0.15 HH HH + H HH + H H + HH HH + H H 0.1 HH H + H HH + H HHH + HH HH +H HH 0.05 HH + HH ++--+-+--+-+--+-+--+-+-+--+-+--+-+--+-+-+--+-+--+-+--+-+-+--+-+--+-+--+-+-**- 7 8 9 10 11 12 13 --------------------------------------------- The occupants of cell, [1, 8], in a standard Young tableau of shape, [n, n], are all the integers from, 8, to , 15 as n goes to infinity, the distribution is [[8., .3515625000e-1], [9., .1093750000], [10., .1845703125], [11., .2197265625 ], [12., .2014160156], [13., .1450195312], [14., .7855224609e-1], [15., .\ 2618408203e-1]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [11.32305908, 1.655753321, .1010601934, 2.401483078] Here is a plot + HHHHH + HHHH HHHH 0.2 HHH HHHH + HHH HH + HH HH + H HH + H HH 0.15 H HH + H HH + HH H + H HH + HH HH 0.1 H H + HH HH + H H | HH HH + H HHH + HH HH 0.05HH HH +H HH ++--+-+-+-+-+-+-+-+-+-+-+--+-+-+-+-+-+-+-+-+-+-+-+--+-+-+-+-+-+-+-+-+-+-+-**- 8 9 10 11 12 13 14 15 --------------------------------------------- The occupants of cell, [1, 9], in a standard Young tableau of shape, [n, n], are all the integers from, 9, to , 17 as n goes to infinity, the distribution is [[9., .1953125000e-1], [10., .7031250000e-1], [11., .1367187500], [12., .187988\ 2812], [13., .2014160156], [14., .1745605469], [15., .1221923828], [16., .\ 6546020508e-1], [17., .2182006836e-1]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [12.95211792, 1.810932385, .6011674913e-1, 2.427894451] Here is a plot 0.2 HHHHH + HHHHHH HHHH 0.18 H HHH + HH HH + HH HH 0.16 HH HH + H HH 0.14 HH H + H HH 0.12 HH HH + H H 0.1 H HH + HH HH 0.08 H H + HH H + HH HH 0.06 HH HH + H HH 0.04 H HHH + HH HH 0.02*+---+---+--+---+---+--+---+---+--+---+--+---+---+--+---+---+--+---+---+-*- 10 12 14 16 --------------------------------------------- The occupants of cell, [1, 10], in a standard Young tableau of shape, [n, n], are all the integers from, 10, to , 19 as n goes to infinity, the distribution is [[10., .1074218750e-1], [11., .4394531250e-1], [12., .9667968750e-1], [13., .15\ 03906250], [14., .1832885742], [15., .1832885742], [16., .1527404785], [17., .1\ 047363281], [18., .5564117432e-1], [19., .1854705811e-1]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [14.59972382, 1.958478036, .2635647674e-1, 2.451858418] Here is a plot 0.18 HHHHHHHHHHH + HH HH + HH HH 0.16 HHH HH + HH HH 0.14 H HH + H H 0.12 HH HH + H HH + H H 0.1 H HH + HH H 0.08 H HH + HH H 0.06 H H + H HH + H HH 0.04 HHH HH + H HH 0.02 HHH HH +**--+--+--+--+---+--+--+--+---+--+--+--+---+--+--+--+---+--+--+--+---+--+--- 10 12 14 16 18 --------------------------------------------- The occupants of cell, [1, 11], in a standard Young tableau of shape, [n, n], are all the integers from, 11, to , 21 as n goes to infinity, the distribution is [[11., .5859375000e-2], [12., .2685546875e-1], [13., .6591796875e-1], [14., .11\ 42578125], [15., .1555175781], [16., .1749572754], [17., .1666259766], [18., .1\ 348876953], [19., .9104919434e-1], [20., .4805374146e-1], [21., .1601791382e-1] ] The limiting average, standard deviation up to the, 4, -th scaled-moment are [16.26334763, 2.099418411, -.2077728314e-2, 2.473637620] Here is a plot + HHHHHH + HHHH HHHH 0.16 HHH HH + HH HH 0.14 HH HH + H HH 0.12 HH H + HH HH + H H 0.1 HH HH + H H 0.08 HH HH + H HH + HH H 0.06 H HH + HH HH 0.04 HH HH + H HH 0.02 HHHH HHH + HHHH H +**+--+--+--+--+--+-+--+--+--+--+--+--+--+--+--+--+--+--+-+--+--+--+--+--+--+ 12 14 16 18 20 --------------------------------------------- The occupants of cell, [1, 12], in a standard Young tableau of shape, [n, n], are all the integers from, 12, to , 23 as n goes to infinity, the distribution is [[12., .3173828125e-2], [13., .1611328125e-1], [14., .4364013672e-1], [15., .\ 8331298828e-1], [16., .1249694824], [17., .1555175781], [18., .1652374268], [19\ ., .1517486572], [20., .1201343536], [21., .8008956909e-1], [22., .4204702377e-\ 1], [23., .1401567459e-1]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [17.94098711, 2.234568941, -.2643665953e-1, 2.493493689] Here is a plot + HHHHHH 0.16 HHHH HHHH + HH HH 0.14 HH H + HHH HH + H HH 0.12 HH HH + H H 0.1 HH HH + HH H + H H 0.08 H HH + H HH 0.06 H H + HH H 0.04 HH HH + HH HH + H HHH 0.02 HHH HH + HHHH H +***+--+-+--+--+-+--+--+-+--+--+-+--+--+-+--+--+-+--+--+-+--+--+-+--+--+-+--+ 0 12 14 16 18 20 22 --------------------------------------------- The occupants of cell, [1, 13], in a standard Young tableau of shape, [n, n], are all the integers from, 13, to , 25 as n goes to infinity, the distribution is [[13., .1708984375e-2], [14., .9521484375e-2], [15., .2819824219e-1], [16., .\ 5874633789e-1], [17., .9613037109e-1], [18., .1307373047], [19., .1525268555], [20., .1552505493], [21., .1386165619], [22., .1078128815], [23., .7115650177e-\ 1], [24., .3719544411e-1], [25., .1239848137e-1]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [19.63102508, 2.364589562, -.4759713080e-1, 2.511663984] Here is a plot + HHHHHHHH + HH HHH 0.14 HH HHH + HH HH + HH HH 0.12 H H + H HH 0.1 H H + HH HH + H H 0.08 HH H + H H 0.06 H HH + HH H + HH HH 0.04 HH H + H HH + HHH HH 0.02 HHH HH + HHHH H +*****--+-+-+--+-+--+-+--+-+-+--+-+--+-+--+-+--+-+-+--+-+--+-+--+-+-+--+-+--+ 0 14 16 18 20 22 24 --------------------------------------------- The occupants of cell, [1, 14], in a standard Young tableau of shape, [n, n], are all the integers from, 14, to , 27 as n goes to infinity, the distribution is [[14., .9155273438e-3], [15., .5554199219e-2], [16., .1785278320e-1], [17., .\ 4028320312e-1], [18., .7133483887e-1], [19., .1050567627], [20., .1330718994], [21., .1478576660], [22., .1455473900], [23., .1270651817], [24., .9741663933e-\ 1], [25., .6376361847e-1], [26., .3321021795e-1], [27., .1107007265e-1]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [21.33213311, 2.490023446, -.6619411844e-1, 2.528355976] Here is a plot + HHHHHHH 0.14 HHH HHH + HH HH + HH HH 0.12 H H + HH HH + HH HH 0.1 H H + HH HH 0.08 H H + HH H + H HH 0.06 H H + HH HH + H HH 0.04 HH H + H HH 0.02 HH HH + HHHH HH + HHH H +******-+-+--+-+-+-+--+-+-+-+--+-+-+-+--+-+-+-+--+-+-+-+--+-+-+-+--+-+-+-+--+ 0 14 16 18 20 22 24 26 --------------------------------------------- The occupants of cell, [1, 15], in a standard Young tableau of shape, [n, n], are all the integers from, 15, to , 29 as n goes to infinity, the distribution is [[15., .4882812500e-3], [16., .3204345703e-2], [17., .1110839844e-1], [18., .\ 2697753906e-1], [19., .5136108398e-1], [20., .8132171631e-1], [21., .1108932495 ], [22., .1330718994], [23., .1423130035], [24., .1363832951], [25., .116899967\ 2], [26., .8856058121e-1], [27., .5756437778e-1], [28., .2988919616e-1], [29., .9963065386e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [23.04320422, 2.611324097, -.8270004828e-1, 2.543748210] Here is a plot 0.14 HHHHHH + HHHH HHH + HH HH 0.12 H HH + HH HH + H H 0.1 H HH + HH H + H HH 0.08 HH H + HH H + H HH 0.06 HH H + HH HH 0.04 H H + HH H + HH HH 0.02 HH HH + HH HH + HHHHH H +*******-+-+-+-+-+-+-+-+-+--+-+-+-+-+-+-+-+-+-+-+--+-+-+-+-+-+-+-+-+-+-+-+--+ 0 16 18 20 22 24 26 28 --------------------------------------------- The occupants of cell, [1, 16], in a standard Young tableau of shape, [n, n], are all the integers from, 16, to , 31 as n goes to infinity, the distribution is [[16., .2593994141e-3], [17., .1831054688e-2], [18., .6809234619e-2], [19., .\ 1770401001e-1], [20., .3604030609e-1], [21., .6099128723e-1], [22., .8894562721\ e-1], [23., .1143586636], [24., .1315124631], [25., .1363832951], [26., .127859\ 3391], [27., .1079332083], [28., .8094990626e-1], [29., .5230609328e-1], [30., .2708708402e-1], [31., .9029028006e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [24.76330435, 2.728874827, -.9747450761e-1, 2.557993459] Here is a plot + HHHHHH + HHH HHH 0.12 HH HH + HH H + HH HH 0.1 H HH + HH H + H HH 0.08 HH H + H H + H H 0.06 H H + HH HH + H H 0.04 HH HH + HH HH + H H 0.02 HHH HHH + HH HH + HHHHH H +********+-+-+-+-+-+-+-+-+-+-+-+-+-+-++-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-++- 0 16 18 20 22 24 26 28 30 --------------------------------------------- The occupants of cell, [1, 17], in a standard Young tableau of shape, [n, n], are all the integers from, 17, to , 33 as n goes to infinity, the distribution is [[17., .1373291016e-3], [18., .1037597656e-2], [19., .4119873047e-2], [20., .\ 1141548157e-1], [21., .2473354340e-1], [22., .4452037811e-1], [23., .6906366348\ e-1], [24., .9455144405e-1], [25., .1160404086], [26., .1289337873], [27., .130\ 3663850], [28., .1199963316], [29., .9999694303e-1], [30., .7435670122e-1], [31\ ., .4780073650e-1], [32., .2469704719e-1], [33., .8232349064e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [26.49163683, 2.843003108, -.1107965844, 2.571222346] Here is a plot + HHHHHHH + HHH HHH 0.12 HHH HH + HH H + HH H 0.1 HH HH + H H + H H 0.08 H H + HH HH + H H 0.06 H H + H H + HH HH 0.04 HH HH + H H + HH HH 0.02 HHH H + HHH HHH + HHHH H +**********-++-+-+-+-++-+-+-+-+-++-+-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-+-++- 0 18 20 22 24 26 28 30 32 --------------------------------------------- The occupants of cell, [1, 18], in a standard Young tableau of shape, [n, n], are all the integers from, 18, to , 35 as n goes to infinity, the distribution is [[18., .7247924805e-4], [19., .5836486816e-3], [20., .2464294434e-2], [21., .\ 7247924805e-2], [22., .1664757729e-1], [23., .3174138069e-1], [24., .5214655399\ e-1], [25., .7564115524e-1], [26., .9849108756e-1], [27., .1163985580], [28., .\ 1257104427], [29., .1244406402], [30., .1127743302], [31., .9294587653e-1], [32\ ., .6860290887e-1], [33., .4390586168e-1], [34., .2263895993e-1], [35., .\ 7546319976e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [28.22751563, 2.953991350, -.1228865740, 2.583546756] Here is a plot + HHHHHH 0.12 HHHH HH + H HH + H HH 0.1 HH H + H HH + H HH 0.08 HH H + H HH + HH H 0.06 H H + H HH + HH H + H H 0.04 HH HH + HH H + HH HH 0.02 HH HH + HHH HHH + HHHH H +***********++-+-+-++-+-+-++-+-++-+-+-++-+-++-+-+-++-+-++-+-+-++-+-++-+-+-++- 0 18 20 22 24 26 28 30 32 34 --------------------------------------------- The occupants of cell, [1, 19], in a standard Young tableau of shape, [n, n], are all the integers from, 19, to , 37 as n goes to infinity, the distribution is [[19., .3814697266e-4], [20., .3261566162e-3], [21., .1459121704e-2], [22., .\ 4539489746e-2], [23., .1101493835e-1], [24., .2216756344e-1], [25., .3842377663\ e-1], [26., .5881190300e-1], [27., .8086636662e-1], [28., .1010829583], [29., .\ 1157859340], [30., .1221015304], [31., .1187098213], [32., .1061539748], [33., .8665630594e-1], [34., .6354795769e-1], [35., .4051182303e-1], [36., .\ 2085167362e-1], [37., .6950557872e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [29.97034499, 3.062085136, -.1339209490, 2.595062832] Here is a plot 0.12 HHHHHH + HHHH HHH + H H + HH HH 0.1 H H + H H + H H 0.08 HH HH + H H + H H 0.06 HH HH + H H + HH H 0.04 H H + HH H + HH HH + HH H 0.02 HH HHH + HHH HH + HHHH HH +************-++-+-++-++-+-++-++-+-++-+-++-++-+-++-+-++-++-+-++-+-++-++-+-++- 0 20 22 24 26 28 30 32 34 36 --------------------------------------------- The occupants of cell, [1, 20], in a standard Young tableau of shape, [n, n], are all the integers from, 20, to , 39 as n goes to infinity, the distribution is [[20., .2002716064e-4], [21., .1811981201e-3], [22., .8561611176e-3], [23., .\ 2808809280e-2], [24., .7178068161e-2], [25., .1520061493e-1], [26., .2770945430\ e-1], [27., .4459902644e-1], [28., .6450930610e-1], [29., .8490968496e-1], [30. , .1025992027], [31., .1144701848], [32., .1182858576], [33., .1132309064], [34\ ., .1000880334], [35., .8102364605e-1], [36., .5907974191e-1], [37., .\ 3753301251e-1], [38., .1928779810e-1], [39., .6429266032e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [31.71960362, 3.167499609, -.1440429278, 2.605853510] Here is a plot 0.12 HHHHHH + HHH HHH + H HH 0.1 HH H + H HH + H H 0.08 HH HH + H H + HH H + H HH 0.06 H H + H HH + HH H 0.04 H H + H HH + H H + HH HH 0.02 HH H + HHH HH + HHHHH HH +*************++-++-++-+-++-++-++-++-++-++-+-++-++-++-++-++-++-+-++-++-++-++- 0 20 22 24 26 28 30 32 34 36 38 --------------------------------------------- The occupants of cell, [1, 21], in a standard Young tableau of shape, [n, n], are all the integers from, 21, to , 41 as n goes to infinity, the distribution is [[21., .1049041748e-4], [22., .1001358032e-3], [23., .4982948303e-3], [24., .\ 1719117165e-2], [25., .4614472389e-2], [26., .1025438309e-1], [27., .1960396767\ e-1], [28., .3308169544e-1], [29., .5017390475e-1], [30., .6928777322e-1], [31. , .8794217370e-1], [32., .1032654312], [33., .1126531977], [34., .1143863238], [35., .1080315281], [36., .9452758706e-1], [37., .7595966817e-1], [38., .\ 5510799456e-1], [39., .3490172989e-1], [40., .1791009823e-1], [41., .5970032744\ e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [33.47483227, 3.270424526, -.1533700931, 2.615990641] Here is a plot + HHHHHH + HH HHH 0.1 HHH H + H HH + H H + HH H 0.08 H H + HH HH + H H + H H 0.06 HH HH + H H + HH H 0.04 H H + HH H + H HH + HH HH 0.02 HH HH + HHH HH + HHHH HH +**************-+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+- 0 25 30 35 40 --------------------------------------------- The occupants of cell, [1, 22], in a standard Young tableau of shape, [n, n], are all the integers from, 22, to , 43 as n goes to infinity, the distribution is [[22., .5483627319e-5], [23., .5507469177e-4], [24., .2878904343e-3], [25., .\ 1041889191e-2], [26., .2930313349e-2], [27., .6816834211e-2], [28., .1363366842\ e-1], [29., .2405941486e-1], [30., .3815672826e-1], [31., .5511527415e-1], [32. , .7322457852e-1], [33., .9012255818e-1], [34., .1032654312], [35., .1104867901 ], [36., .1104867901], [37., .1031210041], [38., .8942524571e-1], [39., .\ 7138990204e-1], [40., .5155937370e-1], [41., .3256381497e-1], [42., .1668895517\ e-1], [43., .5562985057e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [35.23562392, 3.371028244, -.1619999839, 2.625536737] Here is a plot + HHHHHHHH + HHH HHH 0.1 HH H + H HH + HH HH + H H 0.08 HH HH + H H + H H 0.06 H HH + H H + HH HH + HH H 0.04 H HH + H H + H H 0.02 HHH HH + HH HH + HHH HH + HHHH HH +***************---+--+---+--+---+--+---+--+---+--+---+--+---+--+---+--+---+- 0 25 30 35 40 --------------------------------------------- The occupants of cell, [1, 23], in a standard Young tableau of shape, [n, n], are all the integers from, 23, to , 45 as n goes to infinity, the distribution is [[23., .2861022949e-5], [24., .3015995026e-4], [25., .1652240753e-3], [26., .\ 6258487701e-3], [27., .1840293407e-2], [28., .4471912980e-2], [29., .9336099029\ e-2], [30., .1719027758e-1], [31., .2843979746e-1], [32., .4285719479e-1], [33. , .5942864344e-1], [34., .7640825585e-1], [35., .9159194771e-1], [36., .1027473\ 772], [37., .1080849033], [38., .1066437713], [39., .9849737209e-1], [40., .\ 8473670981e-1], [41., .6725135700e-1], [42., .4837378310e-1], [43., .3047548335\ e-1], [44., .1560054505e-1], [45., .5200181683e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [37.00161574, 3.469460967, -.1700142694, 2.634546428] Here is a plot + HHHHHH 0.1 HH HHH + H H + HH H + H HH 0.08 H H + H HH + HH H + H HH 0.06 H H + HH HH + HH H + H H 0.04 HH H + H HH + HH H 0.02 H H + HHH H + HHH HH + HHHHHH HH +****************-+--+---+--+--+---+--+--+---+--+--+---+--+--+---+--+--+---+- 0 25 30 35 40 45 --------------------------------------------- The occupants of cell, [1, 24], in a standard Young tableau of shape, [n, n], are all the integers from, 24, to , 47 as n goes to infinity, the distribution is [[24., .1490116119e-5], [25., .1645088196e-4], [26., .9424984455e-4], [27., .\ 3729015589e-3], [28., .1144129783e-2], [29., .2898462117e-2], [30., .6304155104\ e-2], [31., .1208691392e-1], [32., .2081635175e-1], [33., .3265310079e-1], [34. , .4714291426e-1], [35., .6314293365e-1], [36., .7892866706e-1], [37., .\ 9247263952e-1], [38., .1018299899], [39., .1055328987], [40., .1028945762], [41\ ., .9415189979e-1], [42., .8042141441e-1], [43., .6349059032e-1], [44., .\ 4550158973e-1], [45., .2860099926e-1], [46., .1462551098e-1], [47., .4875170328\ e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [38.77248273, 3.565857371, -.1774819094, 2.643067650] Here is a plot + HHHHHH 0.1 HH HH + HH H + H HH + HH HH 0.08 H H + HH HH + HH H + H H 0.06 H H + HH H + H HH 0.04 H H + H HH + HH H + H HH 0.02 HH H + HHH H + HHH HH + HHHHH HH +*****************---+--+--+--+--+--+---+--+--+--+--+--+---+--+--+--+--+---+- 0 25 30 35 40 45 --------------------------------------------- The occupants of cell, [1, 25], in a standard Young tableau of shape, [n, n], are all the integers from, 25, to , 49 as n goes to infinity, the distribution is [[25., .7748603821e-6], [26., .8940696716e-5], [27., .5346536636e-4], [28., .\ 2205446362e-3], [29., .7047839463e-3], [30., .1858066767e-2], [31., .4202770069\ e-2], [32., .8375520352e-2], [33., .1498777326e-1], [34., .2442451939e-1], [35. , .3663677908e-1], [36., .5100006179e-1], [37., .6630008033e-1], [38., .\ 8087152656e-1], [39., .9286895083e-1], [40., .1006080301], [41., .1028945762], [42., .9926300292e-1], [43., .9007198414e-1], [44., .7644267075e-1], [45., .\ 6006209844e-1], [46., .4290149889e-1], [47., .2691094021e-1], [48., .1374798033\ e-1], [49., .4582660109e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [40.54793239, 3.660338758, -.1844615801, 2.651142644] Here is a plot + HHHHHH 0.1 H HH + HH HH + H H 0.08 HH HH + H H + HH H + H H 0.06 H H + H H + H H + HH H 0.04 H HH + HH H + H H + HH HH 0.02 HH H + HHH H + HH HH + HHHHH HH +*****************--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+---+- 0 25 30 35 40 45 --------------------------------------------- The occupants of cell, [1, 26], in a standard Young tableau of shape, [n, n], are all the integers from, 26, to , 51 as n goes to infinity, the distribution is [[26., .4023313522e-6], [27., .4842877388e-5], [28., .3017485142e-4], [29., .\ 1295506954e-3], [30., .4304861650e-3], [31., .1179157756e-2], [32., .2769234125\ e-2], [33., .5726851523e-2], [34., .1063046814e-1], [35., .1796611282e-1], [36. , .2794728661e-1], [37., .4035169724e-1], [38., .5443275826e-1], [39., .\ 6894816046e-1], [40., .8231566096e-1], [41., .9286895083e-1], [42., .9915695271\ e-1], [43., .1002174549], [44., .9576334577e-1], [45., .8624301315e-1], [46., .\ 7276754235e-1], [47., .5692698891e-1], [48., .4053891634e-1], [49., .2538088675\ e-1], [50., .1295482761e-1], [51., .4318275872e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [42.32770032, 3.753014855, -.1910035579, 2.658808776] Here is a plot 0.1 HHHHHHH + HHH HHH + HH H + H HH 0.08 H H + H HH + HH H + H HH 0.06 HH H + H HH + H H + HH H 0.04 H H + HH HH + H H + HH H 0.02 H HH + HH HH + HHHH HH + HHHHH HH +******************+--+--+--+--+--+--+-+--+--+--+--+--+--+--+--+--+--+--+--+- 0 30 35 40 45 50 --------------------------------------------- The occupants of cell, [1, 27], in a standard Young tableau of shape, [n, n], are all the integers from, 27, to , 53 as n goes to infinity, the distribution is [[27., .2086162567e-6], [28., .2615153790e-5], [29., .1695007086e-4], [30., .\ 7562339306e-4], [31., .2609007061e-3], [32., .7413928397e-3], [33., .1805130392\ e-2], [34., .3868136555e-2], [35., .7436953019e-2], [36., .1301466778e-1], [37. , .2096046496e-1], [38., .3133483650e-1], [39., .4377660982e-1], [40., .\ 5745680039e-1], [41., .7113699095e-1], [42., .8333190369e-1], [43., .9254648919\ e-1], [44., .9753674106e-1], [45., .9753674106e-1], [46., .9240322838e-1], [47. , .8264955427e-1], [48., .6936659019e-1], [49., .5405188846e-1], [50., .\ 3838467441e-1], [51., .2399042151e-1], [52., .1223511497e-1], [53., .4078371656\ e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [44.11154662, 3.843985287, -.1971511989, 2.666099225] Here is a plot + HHHHHH + HH HH + H H + H HH 0.08 HH HH + H H + H H + H HH 0.06 HH H + H H + HH H + H H 0.04 H H + H H + HH HH + H H 0.02 HH H + HH H + HH HH + HHHHH HH +*******************-+--+--+--+-+--+--+--+--+-+--+--+--+--+-+--+--+--+--+--+- 0 30 35 40 45 50 --------------------------------------------- The occupants of cell, [1, 28], in a standard Young tableau of shape, [n, n], are all the integers from, 28, to , 55 as n goes to infinity, the distribution is [[28., .1080334187e-6], [29., .1408159733e-5], [30., .9479932487e-5], [31., .\ 4388857633e-4], [32., .1569860615e-3], [33., .4621669650e-3], [34., .1165045891\ e-2], [35., .2583362628e-2], [36., .5137368862e-2], [37., .9296191274e-2], [38. , .1547815847e-1], [39., .2392079036e-1], [40., .3455225275e-1], [41., .\ 4690351052e-1], [42., .6009512285e-1], [43., .7291541573e-1], [44., .8398293419\ e-1], [45., .9196321300e-1], [46., .9579501354e-1], [47., .9487831485e-1], [48. , .8918561596e-1], [49., .7927610307e-1], [50., .6621356336e-1], [51., .\ 5140804609e-1], [52., .3641403265e-1], [53., .2272235637e-1], [54., .1157966238\ e-1], [55., .3859887461e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [45.89925281, 3.933340855, -.2029421132, 2.673043556] Here is a plot + HHHHHH + HH HHH + HH H 0.08 H HH + H H + HH H + H H 0.06 HH H + H HH + H H + H H + H H 0.04 H H + HH H + HH H + H HH 0.02 HH H + HHH H + HH HH + HHHH HH +********************--+--+--+-+--+--+-+--+--+-+--+--+--+-+--+--+-+--+--+--+- 0 30 35 40 45 50 55 --------------------------------------------- The occupants of cell, [1, 29], in a standard Young tableau of shape, [n, n], are all the integers from, 29, to , 57 as n goes to infinity, the distribution is [[29., .5587935448e-7], [30., .7562339306e-6], [31., .5280598998e-5], [32., .\ 2533430234e-4], [33., .9383074939e-4], [34., .2858228981e-3], [35., .7450450212\ e-3], [36., .1707394840e-2], [37., .3507582878e-2], [38., .6554574065e-2], [39. , .1126762494e-1], [40., .1797698343e-1], [41., .2680778231e-1], [42., .\ 3757672050e-1], [43., .4973389478e-1], [44., .6237459303e-1], [45., .7432972336\ e-1], [46., .8432363575e-1], [47., .9117042668e-1], [48., .9396951873e-1], [49. , .9226098203e-1], [50., .8611024989e-1], [51., .7610754410e-1], [52., .\ 6328507743e-1], [53., .4897059563e-1], [54., .3460588758e-1], [55., .2156212995\ e-1], [56., .1098071433e-1], [57., .3660238109e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [47.69061924, 4.021164572, -.2084091046, 2.679668207] Here is a plot + HHHHHH + H H + HH HH 0.08 H H + HH HH + HH H + H H 0.06 HH HH + H H + HH H + H HH 0.04 HH H + H H + H H + HH H + HH HH 0.02 HH H + HH H + HHH HH + HHHHH HH +*********************+--+--+-+--+-+--+--+-+--+-+--+--+-+--+-+--+--+-+--+--+- 0 30 35 40 45 50 55 --------------------------------------------- The occupants of cell, [1, 30], in a standard Young tableau of shape, [n, n], are all the integers from, 30, to , 59 as n goes to infinity, the distribution is [[30., .2887099981e-7], [31., .4051253200e-6], [32., .2930406481e-5], [33., .\ 1455098391e-4], [34., .5573546514e-4], [35., .1754635014e-3], [36., .4724017344\ e-3], [37., .1117567532e-2], [38., .2369010341e-2], [39., .4566353266e-2], [40. , .8094898971e-2], [41., .1331628402e-1], [42., .2047378668e-1], [43., .\ 2959166739e-1], [44., .4039497453e-1], [45., .5227584940e-1], [46., .6432379906\ e-1], [47., .7542280753e-1], [48., .8440171319e-1], [49., .9021073798e-1], [50. , .9209012836e-1], [51., .8969817697e-1], [52., .8317467319e-1], [53., .\ 7312942280e-1], [54., .6056030326e-1], [55., .4671794823e-1], [56., .3294214298\ e-1], [57., .2049733341e-1], [58., .1043167861e-1], [59., .3477226204e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [49.48546289, 4.107532580, -.2135809292, 2.685996896] Here is a plot + HHHHHH + H HH + HH HH 0.08 HH HH + HH H + H HH + H H 0.06 H H + H H + H H + H H 0.04 H HH + HH H + H H + HH H + H H 0.02 HHH HH + HH H + HHH HH + HHHH HHH +**********************-+--+-+--+-+--+-+--+-+--+-+--+-+--+-+--+-+--+-+--+--+- 0 30 35 40 45 50 55 --------------------------------------------- The occupants of cell, [1, 31], in a standard Young tableau of shape, [n, n], are all the integers from, 31, to , 61 as n goes to infinity, the distribution is [[31., .1490116119e-7], [32., .2165324986e-6], [33., .1620501280e-5], [34., .\ 8318573236e-5], [35., .3291573375e-4], [36., .1069761347e-3], [37., .2971559297\ e-3], [38., .7249298505e-3], [39., .1583971723e-2], [40., .3145943840e-2], [41. , .5744767011e-2], [42., .9732869730e-2], [43., .1541037707e-1], [44., .\ 2293775357e-1], [45., .3225082644e-1], [46., .4300110192e-1], [47., .5454183884\ e-1], [48., .6597156242e-1], [49., .7623380546e-1], [50., .8425841656e-1], [51. , .8911947905e-1], [52., .9018042523e-1], [53., .8719925415e-1], [54., .\ 8037496470e-1], [55., .7032809411e-1], [56., .5802067764e-1], [57., .4463129049\ e-1], [58., .3140720442e-1], [59., .1951733418e-1], [60., .9926919969e-2], [61. , .3308973323e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [51.28361552, 4.192514912, -.2184829137, 2.692050962] Here is a plot + HHHHH + HH HH 0.08 H H + HH HH + H HH + HH H + H HH 0.06 HH H + H H + H H + HH H 0.04 H H + HH HH + HH H + H H + HH H 0.02 H H + HHH H + HHH HH + HHHH HHH +**********************+--+-+--+-+--+-+-+--+-+--+-+--+-+-+--+-+--+-+--+-+--+- 0 35 40 45 50 55 60 --------------------------------------------- The occupants of cell, [1, 32], in a standard Young tableau of shape, [n, n], are all the integers from, 32, to , 63 as n goes to infinity, the distribution is [[32., .7683411241e-8], [33., .1154839993e-6], [34., .8931965567e-6], [35., .\ 4734902177e-5], [36., .1933418389e-4], [37., .6480285083e-4], [38., .1855367336\ e-3], [39., .4662960237e-3], [40., .1049166053e-2], [41., .2144961709e-2], [42. , .4030740545e-2], [43., .7025915336e-2], [44., .1144372574e-1], [45., .\ 1752189509e-1], [46., .2534416968e-1], [47., .3477042226e-1], [48., .4539471795\ e-1], [49., .5654705350e-1], [50., .6734596997e-1], [51., .7679803593e-1], [52. , .8392928212e-1], [53., .8792591460e-1], [54., .8825896731e-1], [55., .\ 8477047058e-1], [56., .7770626470e-1], [57., .6769079058e-1], [58., .5564964033\ e-1], [59., .4269416851e-1], [60., .2998757074e-1], [61., .1861297494e-1], [62. , .9461595595e-2], [63., .3153865198e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [53.08492201, 4.276176165, -.2231374618, 2.697849669] Here is a plot + HHHHH + HH HH 0.08 H HH + HH H + H HH + H HH 0.06 HH H + H HH + HH H + H H + H HH 0.04 H H + H HH + H H + HH H 0.02 HH H + H H + HH H + HHH HH + HHHHH HHH +***********************-+-+--+-+-+--+-+--+-+-+--+-+-+--+-+-+--+-+-+--+-+--+- 0 35 40 45 50 55 60 --------------------------------------------- The occupants of cell, [1, 33], in a standard Young tableau of shape, [n, n], are all the integers from, 33, to , 65 as n goes to infinity, the distribution is [[33., .3958120942e-8], [34., .6146728992e-7], [35., .4908069968e-6], [36., .\ 2684100764e-5], [37., .1129919838e-4], [38., .3901989840e-4], [39., .1150414246\ e-3], [40., .2975816442e-3], [41., .6888463986e-3], [42., .1448343710e-2], [43. , .2798200047e-2], [44., .5013441752e-2], [45., .8392065541e-2], [46., .\ 1320429893e-1], [47., .1962679807e-1], [48., .2767378527e-1], [49., .3714113287\ e-1], [50., .4757949047e-1], [51., .5830819911e-1], [52., .6847377329e-1], [53. , .7714711791e-1], [54., .8344484182e-1], [55., .8665425881e-1], [56., .\ 8634029411e-1], [57., .8241573528e-1], [58., .7516315058e-1], [59., .6520563918\ e-1], [60., .5343239877e-1], [61., .4089214192e-1], [62., .2867150180e-1], [63. , .1777633112e-1], [64., .9031523068e-2], [65., .3010507689e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [54.88923901, 4.358576068, -.2275644728, 2.703410446] Here is a plot + HHHHH + HH HHH 0.08 HH HH + H H + HH H + H H 0.06 H HH + HH H + H HH + H H + H H 0.04 HH HH + H H + HH HH + HH H 0.02 HHH H + HH H + HH H + HHH HH + HHHH HHH +************************--+-+-+-+--+-+-+--+-+-+-+--+-+-+--+-+-+-+--+-+-+--+- 0 35 40 45 50 55 60 65 --------------------------------------------- The occupants of cell, [1, 34], in a standard Young tableau of shape, [n, n], are all the integers from, 34, to , 67 as n goes to infinity, the distribution is [[34., .2037268132e-8], [35., .3265449777e-7], [36., .2689193934e-6], [37., .\ 1515727490e-5], [38., .6572099664e-5], [39., .2336275429e-4], [40., .7086702135\ e-4], [41., .1885132588e-3], [42., .4485605666e-3], [43., .9691123353e-3], [44. , .1923315250e-2], [45., .3538900060e-2], [46., .6082484478e-2], [47., .\ 9825551849e-2], [48., .1499353691e-1], [49., .2170492963e-1], [50., .2991210614\ e-1], [51., .3935803440e-1], [52., .4956196924e-1], [53., .5984262540e-1], [54. , .6938004382e-1], [55., .7730919169e-1], [56., .8283127681e-1], [57., .\ 8532452594e-1], [58., .8443572880e-1], [59., .8013718260e-1], [60., .7273990421\ e-1], [61., .6286164561e-1], [62., .5135571941e-1], [63., .3921249511e-1], [64. , .2744874658e-1], [65., .1700051401e-1], [66., .8633073521e-2], [67., .\ 2877691174e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [56.69643370, 4.439769984, -.2317816886, 2.708749113] Here is a plot + HHHHH 0.08 HH HHH + HH H + H H + HH H + H H 0.06 H H + HH HH + HH H + H HH + H H 0.04 H H + H HH + H H + HH HH 0.02 HH H + HH H + HH HH + HH HH + HHHHH HHH +************************-+-+-+-+--+-+-+-+--+-+-+-+-+--+-+-+-+-+--+-+-+-+--+- 0 35 40 45 50 55 60 65 --------------------------------------------- The occupants of cell, [1, 35], in a standard Young tableau of shape, [n, n], are all the integers from, 35, to , 69 as n goes to infinity, the distribution is [[35., .1047737896e-8], [36., .1731677912e-7], [37., .1469452400e-6], [38., .\ 8528586477e-6], [39., .3805558663e-5], [40., .1391407386e-4], [41., .4338797226\ e-4], [42., .1185937908e-3], [43., .2898391353e-3], [44., .6429368121e-3], [45. , .1309686099e-2], [46., .2472833893e-2], [47., .4360430431e-2], [48., .\ 7225456836e-2], [49., .1130941070e-1], [50., .1679276134e-1], [51., .2373976678\ e-1], [52., .3204868515e-1], [53., .4141964572e-1], [54., .5135067189e-1], [55. , .6116771210e-1], [56., .7008800345e-1], [57., .7730919169e-1], [58., .\ 8211100483e-1], [59., .8395323892e-1], [60., .8255401827e-1], [61., .7793561165\ e-1], [62., .7043070090e-1], [63., .6064865911e-1], [64., .4940774384e-1], [65. , .3764399531e-1], [66., .2631031930e-1], [67., .1627951007e-1], [68., .\ 8263084656e-2], [69., .2754361552e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [58.50638276, 4.519809340, -.2358049846, 2.713880060] Here is a plot + HHHHH 0.08 HH HH + HH H + H H + H HH + HH H 0.06 H H + H H + H H + HH H 0.04 H H + H H + HH HH + H H + HH H 0.02 HH H + HH H + H HH + HHH HH + HHHHH HHH +************************+-+-+--+-+-+-+-+-+-+--+-+-+-+-+-+-+--+-+-+-+-+-+--+- 0 35 40 45 50 55 60 65 --------------------------------------------- The occupants of cell, [1, 36], in a standard Young tableau of shape, [n, n], are all the integers from, 36, to , 71 as n goes to infinity, the distribution is [[36., .5384208634e-9], [37., .9167706594e-8], [38., .8009010344e-7], [39., .\ 4782523320e-6], [40., .2194334229e-5], [41., .8245377103e-5], [42., .2641097353\ e-4], [43., .7412111927e-4], [44., .1859204742e-3], [45., .4231293550e-3], [46. , .8840381167e-3], [47., .1711521606e-2], [48., .3093904442e-2], [49., .\ 5254877699e-2], [50., .8429699642e-2], [51., .1282780380e-1], [52., .1858573846\ e-1], [53., .2571808068e-1], [54., .3407645690e-1], [55., .4332712941e-1], [56. , .5295538039e-1], [57., .6230044751e-1], [58., .7061897317e-1], [59., .\ 7716913880e-1], [60., .8130319981e-1], [61., .8255401827e-1], [62., .8070184478\ e-1], [63., .7581082389e-1], [64., .6822974150e-1], [65., .5855732604e-1], [66. , .4757782740e-1], [67., .3617668904e-1], [68., .2524831423e-1], [69., .\ 1560804879e-1], [70., .7918789462e-2], [71., .2639596487e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [60.31897141, 4.598742005, -.2396486137, 2.718816410] Here is a plot + HHHHH 0.08 HHH HH + HH HH + H HH + HH H 0.06 H H + HH H + H H + HH HH + H H 0.04 HH H + H H + HH H + H HH + HH H 0.02 H HH + HHH H + HH HH + HH HH + HHHHH HHH +*************************-+-+-+-+-+-+-+-+-+-+-+-+--+-+-+-+-+-+-+-+-+-+-+--+- 0 40 45 50 55 60 65 70 --------------------------------------------- The occupants of cell, [1, 37], in a standard Young tableau of shape, [n, n], are all the integers from, 37, to , 73 as n goes to infinity, the distribution is [[37., .2764863893e-9], [38., .4845787771e-8], [39., .4354660632e-7], [40., .\ 2673277777e-6], [41., .1260259523e-5], [42., .4863119102e-5], [43., .1598934614\ e-4], [44., .4604075116e-4], [45., .1184435453e-3], [46., .2763682724e-3], [47. , .5918093006e-3], [48., .1174011307e-2], [49., .2174095014e-2], [50., .\ 3782153455e-2], [51., .6213537820e-2], [52., .9682763102e-2], [53., .1436627352\ e-1], [54., .2035862290e-1], [55., .2762955965e-1], [56., .3599113691e-1], [57. , .4508363465e-1], [58., .5438660688e-1], [59., .6325714972e-1], [60., .\ 7099239900e-1], [61., .7690843225e-1], [62., .8042424630e-1], [63., .8113807097\ e-1], [64., .7888423566e-1], [65., .7376188270e-1], [66., .6613134311e-1], [67. , .5657903799e-1], [68., .4585639773e-1], [69., .3480173042e-1], [70., .\ 2425575150e-1], [71., .1498149358e-1], [72., .7597757457e-2], [73., .2532585819\ e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [62.13409264, 4.676612639, -.2433254129, 2.723570158] Here is a plot 0.08 HHHHH + HHH HHH + HH HH + H H + HH HH 0.06 HH H + H H + HH H + H H + H H 0.04 H H + H H + H H + HH HH + H H 0.02 H H + HH H + H H + HH HH + HHHHH HHH +**************************-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+--+- 0 40 45 50 55 60 65 70 --------------------------------------------- The occupants of cell, [1, 38], in a standard Young tableau of shape, [n, n], are all the integers from, 38, to , 75 as n goes to infinity, the distribution is [[38., .1418811735e-9], [39., .2557499101e-8], [40., .2362321538e-7], [41., .\ 1489752322e-6], [42., .7210815056e-6], [43., .2855482762e-5], [44., .9630255590\ e-5], [45., .2843218317e-4], [46., .7496767047e-4], [47., .1792237857e-3], [48. , .3930975033e-3], [49., .7985178123e-3], [50., .1513856686e-2], [51., .\ 2695613757e-2], [52., .4532185767e-2], [53., .7227325569e-2], [54., .1097273648\ e-1], [55., .1591187105e-1], [56., .2209982091e-1], [57., .2946642788e-1], [58. , .3779069375e-1], [59., .4669376446e-1], [60., .5565519400e-1], [61., .\ 6405329233e-1], [62., .7122592663e-1], [63., .7654412915e-1], [64., .7948813412\ e-1], [65., .7971459604e-1], [66., .7710489200e-1], [67., .7178731324e-1], [68. , .6412999983e-1], [69., .5470587799e-1], [70., .4423483103e-1], [71., .\ 3351123563e-1], [72., .2332644833e-1], [73., .1439575097e-1], [74., .7297845978\ e-2], [75., .2432615326e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [63.95164649, 4.753462970, -.2468469783, 2.728152294] Here is a plot 0.08 HHHHH + HH H + HH HH + H HH + H H 0.06 H HH + H H + H H + HH H + H H 0.04 H HH + HH H + H H + HH H + HH H 0.02 H H + HHH HH + HH H + HHH HH + HHHHH HHH +**************************+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+- 0 40 45 50 55 60 65 70 75 --------------------------------------------- The occupants of cell, [1, 39], in a standard Young tableau of shape, [n, n], are all the integers from, 39, to , 77 as n goes to infinity, the distribution is [[39., .7275957614e-10], [40., .1347871148e-8], [41., .1278749551e-7], [42., .\ 8278220776e-7], [43., .4111143426e-6], [44., .1669581025e-5], [45., .5771980113\ e-5], [46., .1746145244e-4], [47., .4715914997e-4], [48., .1154416692e-3], [49. , .2591851670e-3], [50., .5387909835e-3], [51., .1045068718e-2], [52., .\ 1903518022e-2], [53., .3273245276e-2], [54., .5337907681e-2], [55., .8287101674\ e-2], [56., .1228847185e-1], [57., .1745319190e-1], [58., .2379980713e-1], [59. , .3122308031e-1], [60., .3947489439e-1], [61., .4816314866e-1], [62., .\ 5677202064e-1], [63., .6470340588e-1], [64., .7133550498e-1], [65., .7609120531\ e-1], [66., .7850679913e-1], [67., .7829112111e-1], [68., .7536645279e-1], [69. , .6988525623e-1], [70., .6222042167e-1], [71., .5293056705e-1], [72., .\ 4270534387e-1], [73., .3229815923e-1], [74., .2245491070e-1], [75., .1384719493\ e-1], [76., .7017159594e-2], [77., .2339053198e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [65.77153940, 4.829332070, -.2502238150, 2.732572902] Here is a plot 0.08 HHHHH + HH HH + HH HH + H H + H H 0.06 HH H + HH H + H HH + HH H + H H 0.04 H HH + H H + H H + H HH + HH H 0.02 H HH + HH H + HH H + HHH HH + HHHH HHH +**************************-+-+-+-+-+-+-+-+-+-+-+-++-+-+-+-+-+-+-+-+-+-+-+-+- 0 40 45 50 55 60 65 70 75 --------------------------------------------- The occupants of cell, [1, 40], in a standard Young tableau of shape, [n, n], are all the integers from, 40, to , 79 as n goes to infinity, the distribution is [[40., .3728928277e-10], [41., .7094058674e-9], [42., .6907839634e-8], [43., .\ 4587514013e-7], [44., .2336010425e-6], [45., .9722854202e-6], [46., .3443510863\ e-5], [47., .1066785610e-4], [48., .2949348452e-4], [49., .7388266829e-4], [50. , .1696992537e-3], [51., .3607975336e-3], [52., .7155817749e-3], [53., .\ 1332462615e-2], [54., .2342006995e-2], [55., .3903344992e-2], [56., .6192806958\ e-2], [57., .9383923955e-2], [58., .1361972296e-1], [59., .1898034619e-1], [60. , .2545091875e-1], [61., .3289574533e-1], [62., .4104491860e-1], [63., .\ 4949810550e-1], [64., .5774778974e-1], [65., .6522103312e-1], [66., .7133550498\ e-1], [67., .7556279416e-1], [68., .7749041646e-1], [69., .7687378185e-1], [70. , .7367070761e-1], [71., .6805358620e-1], [72., .6039755776e-1], [73., .\ 5124641264e-1], [74., .4126089841e-1], [75., .3115618860e-1], [76., .2163624208\ e-1], [77., .1333260323e-1], [78., .6754016109e-2], [79., .2251338703e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [67.59368364, 4.904256573, -.2534654642, 2.736841261] Here is a plot + HHHHH + H HH + H HH + HH H 0.06 HH H | H H + H HH + H H + H HH + H H 0.04 H H + H HH + HH H + HH H + H HH 0.02 HH H + H H + HH H + HHH HH + HHH HHH +***************************+--+---+---+---+--+---+---+---+--+---+---+---+--- 0 40 50 60 70 ----------------------- This took, 232.602, seconds.