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], 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, n], are all the integers from, 2, to , 4 as n goes to infinity, the distribution is [[2., .6666666667], [3., .2962962963], [4., .3703703704e-1]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [2.370370370, .5543196127, 1.176426717, 3.392697704] Here is a plot +HH + HHH 0.6 HHHH + HHH + HHH 0.5 HHH + HHHH + HHH 0.4 HHH + HHH + HHH + HHHH 0.3 HHHH + HHHH + HHHHH 0.2 HHHHH + HHHH + HHHHH 0.1 HHHH + HHHHH ++---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+-***- 2 2.5 3 3.5 4 --------------------------------------------- The occupants of cell, [1, 3], in a standard Young tableau of shape, [n, n, n], are all the integers from, 3, to , 7 as n goes to infinity, the distribution is [[3., .3703703704], [4., .3703703704], [5., .1975308642], [6., .5486968450e-1], [7., .6858710562e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [3.957475995, .9204890418, .7173355522, 2.907492361] Here is a plot +HHHHHHHHHHHHHHHHHHHHH 0.35 HH + HH + HH 0.3 HH + HH + HH 0.25 H + HH 0.2 HH + HHH + HH 0.15 HHH + HH + HHH 0.1 HH + HH + HHH 0.05 HHHHHHHH + HHHHHHH ++---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---*****- 0 3 4 5 6 7 --------------------------------------------- The occupants of cell, [1, 4], in a standard Young tableau of shape, [n, n, n], are all the integers from, 4, to , 10 as n goes to infinity, the distribution is [[4., .1851851852], [5., .2962962963], [6., .2674897119], [7., .1646090535], [8\ ., .6721536351e-1], [9., .1707056851e-1], [10., .2133821064e-2]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [5.692120104, 1.250397444, .5091144359, 2.763670926] Here is a plot 0.3 HHH + HH HHHHHHH + HH HHHH 0.25 HH H + HH HH + HH HH + HH HH 0.2 H HH +H HH + HH 0.15 HH + H + HH 0.1 H + HH + HH + HHHH 0.05 HHHH + HHH + HHHHHHHHH ++--+-+--+-+--+-+--+-+-+--+-+--+-+--+-+-+--+-+--+-+--+-+-+--+-+--+-+********- 0 4 5 6 7 8 9 10 --------------------------------------------- The occupants of cell, [1, 5], in a standard Young tableau of shape, [n, n, n], are all the integers from, 5, to , 13 as n goes to infinity, the distribution is [[5., .8641975309e-1], [6., .1920438957], [7., .2414266118], [8., .2176497485], [9., .1493674745], [10., .7681755830e-1], [11., .2845094752e-1], [12., .\ 6954676060e-2], [13., .8693345075e-3]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [7.535747148, 1.554848442, .3835846841, 2.707746811] Here is a plot + HHHH + HH HHHHH + HHH HHH 0.2 HH HH + HH HH + H HH + HH H + H HH 0.15 H HH + H HH + HH H + H H 0.1 H HH +H H + HH + HHH 0.05 HH + HH + HHHHH + HHHH +--+---+---+--+---+---+--+---+---+--+---+--+---+---+--+---+---+--***********- 0 6 8 10 12 --------------------------------------------- The occupants of cell, [1, 6], in a standard Young tableau of shape, [n, n, n], are all the integers from, 6, to , 16 as n goes to infinity, the distribution is [[6., .3840877915e-1], [7., .1097393690], [8., .1760402378], [9., .2048468221], [10., .1883859168], [11., .1406289692], [12., .8444963787e-1], [13., .\ 3974100606e-1], [14., .1399214588e-1], [15., .3348547733e-2], [16., .4185684666\ e-3]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [9.464193475, 1.839257186, .2974828100, 2.685177671] Here is a plot 0.2 HHHHH + HHH HHHH + HHH HH + H H + HH H 0.15 H HH + H HH + H H + HH HH + H H 0.1 H H + H H + H HH + H HH + H H 0.05 H HH +HH HH + HHH + HHH + HHHHHH ++--+--+--+--+--+--+--+--+--+--+--+--+-+--+--+--+--+--+--+--+--+--**********- 0 6 8 10 12 14 16 --------------------------------------------- The occupants of cell, [1, 7], in a standard Young tableau of shape, [n, n, n], are all the integers from, 7, to , 19 as n goes to infinity, the distribution is [[7., .1646090535e-1], [8., .5761316872e-1], [9., .1121780216], [10., .15851242\ 19], [11., .1790885536], [12., .1687750851], [13., .1346226580], [14., .\ 9041078878e-1], [15., .5022821599e-1], [16., .2242331071e-1], [17., .7653823389\ e-2], [18., .1807152745e-2], [19., .2258940931e-3]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [11.46111075, 2.107186934, .2338235384, 2.677644659] Here is a plot 0.18 HHHH + HHH HHHH 0.16 HH HH + H H 0.14 H HH + HH H + H HH 0.12 HH H + H H 0.1 H HH + H H 0.08 HH HH + H HH 0.06 H H + HH HH + HH HH 0.04 HH HH + H H 0.02H HHHH + HHHHHH +--+-+--+-+-+--+-+--+-+--+-+-+--+-+--+-+--+-+--+-+-+--+-+--+-+--+-**********+ 0 8 10 12 14 16 18 --------------------------------------------- The occupants of cell, [1, 8], in a standard Young tableau of shape, [n, n, n], are all the integers from, 8, to , 22 as n goes to infinity, the distribution is [[8., .6858710562e-2], [9., .2845094752e-1], [10., .6508154245e-1], [11., .1077\ 071585], [12., .1429227327], [13., .1597919619], [14., .1542723777], [15., .129\ 7960883], [16., .9487551909e-1], [17., .5952973747e-1], [18., .3143692795e-1], [19., .1355364558e-1], [20., .4530431533e-2], [21., .1059750066e-2], [22., .\ 1324687583e-3]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [13.51482079, 2.361215679, .1843701158, 2.677546827] Here is a plot 0.16 HHHHH + HHH HHH 0.14 HH H + H HH + HH HH 0.12 H H + H H 0.1 HH HH + H H + H H 0.08 H HH + H H + H H 0.06 HH HH + H HH 0.04 H HH + HH HHH + HH HH 0.02 HH HHH +HH HHHHH ++--+-+-+-+-+-+-+-+-+-+-+--+-+-+-+-+-+-+-+-+-+-+-+--+-+-+-+-+-+-+***********- 0 8 10 12 14 16 18 20 22 --------------------------------------------- The occupants of cell, [1, 9], in a standard Young tableau of shape, [n, n, n], are all the integers from, 9, to , 25 as n goes to infinity, the distribution is [[9., .2794289488e-2], [10., .1341258954e-1], [11., .3522498264e-1], [12., .\ 6664897891e-1], [13., .1010497873], [14., .1295170427], [15., .1445456438], [16\ ., .1427119153], [17., .1254376611], [18., .9798783771e-1], [19., .6746703580e-\ 1], [20., .4037647752e-1], [21., .2057681379e-1], [22., .8654625541e-2], [23., .2849550178e-2], [24., .6620167081e-3], [25., .8275208851e-4]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [15.61663445, 2.603301207, .1445742114, 2.681305669] Here is a plot + HHHHHH 0.14 HH HHH + HH HH 0.12 H HH + H H + H HH 0.1 H HH + H H + HH H 0.08 H HH + H H + H HH 0.06 H H + HH HH 0.04 H HH + H H + HH H 0.02 H HHH + HHH HH + HHHH HHHHH +*+-+-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-+-++-+-+-+-+-++-+-+-+-+************- 0 10 12 14 16 18 20 22 24 --------------------------------------------- The occupants of cell, [1, 10], in a standard Young tableau of shape, [n, n, n], are all the integers from, 10, to , 28 as n goes to infinity, the distribution is [[10., .1117715795e-2], [11., .6096631611e-2], [12., .1806973869e-1], [13., .\ 3841630586e-1], [14., .6531298865e-1], [15., .9391154323e-1], [16., .1179602042 ], [17., .1319704699], [18., .1329626322], [19., .1212070124], [20., .\ 9988626079e-1], [21., .7398982281e-1], [22., .4876937691e-1], [23., .2818935176\ e-1], [24., .1399262587e-1], [25., .5777600361e-2], [26., .1880729284e-2], [27. , .4346574346e-3], [28., .5433217932e-4]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [17.75987773, 2.834977895, .1116910007, 2.687089999] Here is a plot + HHHHHH + HHH HH 0.12 HH HH + H H + HH HH 0.1 H H + H H + H H 0.08 HH H + H H + H HH 0.06 H H + H H + HH HH 0.04 H H + HH H + H HH 0.02 H HH + HHH HHHH + HHH HHHH +***-++-+-++-++-+-++-+-++-++-+-++-+-++-++-+-++-+-++-++-+-++-+-++-***********- 0 10 12 14 16 18 20 22 24 26 28 --------------------------------------------- The occupants of cell, [1, 11], in a standard Young tableau of shape, [n, n, n], are all the integers from, 11, to , 31 as n goes to infinity, the distribution is [[11., .4403122830e-3], [12., .2690797285e-2], [13., .8881512717e-2], [14., .\ 2093929524e-1], [15., .3937714559e-1], [16., .6256964381e-1], [17., .8692694619\ e-1], [18., .1078815426], [19., .1212435205], [20., .1243953896], [21., .116945\ 0255], [22., .1007083699], [23., .7912800495e-1], [24., .5630835087e-1], [25., .3590209618e-1], [26., .2020701136e-1], [27., .9829185168e-2], [28., .\ 4000824114e-2], [29., .1290800866e-2], [30., .2970890883e-3], [31., .3713613604\ e-4]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [19.93929192, 3.057476705, .8395299924e-1, 2.693907543] Here is a plot + HHHHHH 0.12 HH HH + H H + HH HH 0.1 H H + HH H + H HH 0.08 H H + HH H + H H 0.06 H H + HH H + H H + H H 0.04 HH HH + HH H + H HH 0.02 H HHH + HH HHH + HHHH HHHH +*****---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---************- 0 15 20 25 30 --------------------------------------------- The occupants of cell, [1, 12], in a standard Young tableau of shape, [n, n, n], are all the integers from, 12, to , 34 as n goes to infinity, the distribution is [[12., .1712325545e-3], [13., .1159112677e-2], [14., .4215582413e-2], [15., .\ 1090814792e-1], [16., .2245292969e-1], [17., .3899093731e-1], [18., .5918109303\ e-1], [19., .8032351640e-1], [20., .9897917402e-1], [21., .1118542727], [22., .\ 1166446395], [23., .1125867931], [24., .1005895037], [25., .8296041544e-1], [26\ ., .6281811169e-1], [27., .4331984446e-1], [28., .2690810680e-1], [29., .\ 1483002534e-1], [30., .7098715234e-2], [31., .2856625849e-2], [32., .9150724803\ e-3], [33., .2099090994e-3], [34., .2623863743e-4]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [22.15064564, 3.271803695, .6016413501e-1, 2.701199193] Here is a plot + HHHHH + HH HH + HH H 0.1 H HH + H H + H HH 0.08 H H + H HH + H H + H H 0.06 H H + H HH + HH H 0.04 H HH + H H + H HH + H HH 0.02 HH HH + HH HH + HHHH HHHH +******-+--+---+--+--+---+--+--+---+--+--+---+--+--+---+--+--+---***********- 0 15 20 25 30 --------------------------------------------- The occupants of cell, [1, 13], in a standard Young tableau of shape, [n, n, n], are all the integers from, 13, to , 37 as n goes to infinity, the distribution is [[13., .6585867481e-4], [14., .4892358700e-3], [15., .1943562670e-2], [16., .\ 5473587640e-2], [17., .1222950292e-1], [18., .2301065703e-1], [19., .3780698341\ e-1], [20., .5554572515e-1], [21., .7416768296e-1], [22., .9102487282e-1], [23. , .1034750834], [24., .1094915873], [25., .1081178921], [26., .9966023933e-1], [27., .8559407943e-1], [28., .6822181079e-1], [29., .5015321931e-1], [30., .\ 3372312737e-1], [31., .2050994577e-1], [32., .1111321630e-1], [33., .5250610852\ e-2], [34., .2093324261e-2], [35., .6666343926e-3], [36., .1524980637e-3], [37. , .1906225796e-4]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [24.39047384, 3.478793865, .3948257178e-1, 2.708642097] Here is a plot + HHHHH + HH HH 0.1 HH HH + HH H + H HH 0.08 H H + HH HH + H H + HH HH 0.06 H H + H H + HH H + H HH 0.04 H H + HH HH + H H 0.02 HH HH + HH HH + HHH HH + HHH HHHH +********--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--***********- 0 15 20 25 30 35 --------------------------------------------- The occupants of cell, [1, 14], in a standard Young tableau of shape, [n, n, n], are all the integers from, 14, to , 40 as n goes to infinity, the distribution is [[14., .2508901898e-4], [15., .2029422868e-3], [16., .8742826196e-3], [17., .\ 2661294457e-2], [18., .6410337271e-2], [19., .1297864244e-1], [20., .2291728045\ e-1], [21., .3616489737e-1], [22., .5187587737e-1], [23., .6846166582e-1], [24. , .8384927704e-1], [25., .9588931136e-1], [26., .1028026793], [27., .1035508793 ], [28., .9804412734e-1], [29., .8715033542e-1], [30., .7251246498e-1], [31., .\ 5621086275e-1], [32., .4033487684e-1], [33., .2656388120e-1], [34., .1587754735\ e-1], [35., .8483249428e-2], [36., .3964949656e-2], [37., .1568551512e-2], [38. , .4970710695e-3], [39., .1134456495e-3], [40., .1418070619e-4]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [26.65589646, 3.679149343, .2129656735e-1, 2.716047774] Here is a plot + HHHHHH 0.1 HHH HH + H HH + HH HH + H H 0.08 HH HH + H H + H H 0.06 HH HH + H H + H H + H H 0.04 HH H + H H + H HH + H HH 0.02 HH HH + H HH + HHHH H + HHH HHHH +**********--+--+-+--+--+--+--+-+--+--+--+--+-+--+--+--+--+-+--+-***********- 0 15 20 25 30 35 40 --------------------------------------------- The occupants of cell, [1, 15], in a standard Young tableau of shape, [n, n, n], are all the integers from, 15, to , 43 as n goes to infinity, the distribution is [[15., .9478073835e-5], [16., .8293314606e-4], [17., .3850699801e-3], [18., .\ 1259530701e-2], [19., .3252164085e-2], [20., .7044929418e-2], [21., .1329134087\ e-1], [22., .2239111084e-1], [23., .3427612137e-1], [24., .4828555430e-1], [25. , .6318324498e-1], [26., .7732646948e-1], [27., .8894975640e-1], [28., .\ 9649535335e-1], [29., .9891203934e-1], [30., .9585605802e-1], [31., .8775554608\ e-1], [32., .7573043831e-1], [33., .6138341807e-1], [34., .4649717221e-1], [35. , .3269735492e-1], [36., .2116319110e-1], [37., .1246671671e-1], [38., .\ 6582833082e-2], [39., .3048851830e-2], [40., .1198269673e-2], [41., .3781521650\ e-3], [42., .8613465980e-4], [43., .1076683247e-4]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [28.94448873, 3.873467224, .5149645665e-2, 2.723306673] Here is a plot 0.1 HHHHHH + HH H + H HH + HH HH 0.08 HH H + H H + HH H + H H 0.06 H HH + H H + H H + H HH 0.04 HH H + H HH + HH H + H HH 0.02 HHH HH + HH H + HH HHH + HHHHH HHHH +***********+--+-+--+-+--+--+-+--+-+--+--+-+--+-+--+--+-+--+-+--+***********- 0 15 20 25 30 35 40 --------------------------------------------- The occupants of cell, [1, 16], in a standard Young tableau of shape, [n, n, n], are all the integers from, 16, to , 46 as n goes to infinity, the distribution is [[16., .3554277688e-5], [17., .3345202530e-4], [18., .1665244917e-3], [19., .\ 5823749812e-3], [20., .1604125279e-2], [21., .3700145038e-2], [22., .7422827747\ e-2], [23., .1328284020e-1], [24., .2158512149e-1], [25., .3227351876e-1], [26. , .4483522452e-1], [27., .5830238431e-1], [28., .7136177511e-1], [29., .\ 8255374674e-1], [30., .9051811563e-1], [31., .9423368658e-1], [32., .9320126839\ e-1], [33., .8753494390e-1], [34., .7794645406e-1], [35., .6562571557e-1], [36. , .5203545096e-1], [37., .3865425447e-1], [38., .2672078040e-1], [39., .\ 1704187220e-1], [40., .9915545991e-2], [41., .5183465191e-2], [42., .2382161685\ e-2], [43., .9310143375e-3], [44., .2927628418e-3], [45., .6657139556e-4], [46. , .8321424445e-5]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [31.25418634, 4.062260317, -.9306513665e-2, 2.730356730] Here is a plot + HHHHHH + HH HH + HH H 0.08 H HH + HH H + H H + HH H 0.06 H H + H HH + H H + HH H + H H 0.04 HH H + H H + H HH + HH H 0.02 H H + H HH + HHH HHH + HHHH HHH +************-+-+--+-+-+--+-+--+-+--+-+-+--+-+--+-+--+-+-+--+-+--***********- 0 20 25 30 35 40 45 --------------------------------------------- The occupants of cell, [1, 17], in a standard Young tableau of shape, [n, n, n], are all the integers from, 17, to , 49 as n goes to infinity, the distribution is [[17., .1324142668e-5], [18., .1333951132e-4], [19., .7086873508e-4], [20., .\ 2638476872e-3], [21., .7720532800e-3], [22., .1888536230e-2], [23., .4011971857\ e-2], [24., .7594265709e-2], [25., .1304426175e-1], [26., .2060575795e-1], [27. , .3024114350e-1], [28., .4155546784e-1], [29., .5378809764e-1], [30., .\ 6588284021e-1], [31., .7662774197e-1], [32., .8483855500e-1], [33., .8954963705\ e-1], [34., .9017488665e-1], [35., .8660864820e-1], [36., .7924913882e-1], [37. , .6894027521e-1], [38., .5683938881e-1], [39., .4422985158e-1], [40., .\ 3231107434e-1], [41., .2201018293e-1], [42., .1386094940e-1], [43., .7979427457\ e-2], [44., .4135436452e-2], [45., .1887821193e-2], [46., .7342433334e-3], [47. , .2301716879e-3], [48., .5226120450e-4], [49., .6532650563e-5]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [33.58321456, 4.245972894, -.2234283674e-1, 2.737164947] Here is a plot + HHHHH + H HH 0.08 H HH + HH H + H HH + HH H + H HH 0.06 HH H + H H + H H + H H 0.04 HH H + H H + HH H + H H + HH H 0.02 H H + HH HH + HHH HH + HHHH HHHH +*************-+-+--+-+-+--+-+-+-+--+-+-+--+-+-+-+--+-+-+--+-+-+-***********- 0 20 25 30 35 40 45 --------------------------------------------- The occupants of cell, [1, 18], in a standard Young tableau of shape, [n, n, n], are all the integers from, 18, to , 52 as n goes to infinity, the distribution is [[18., .4904232104e-6], [19., .5265596575e-5], [20., .2973599399e-4], [21., .\ 1174109493e-3], [22., .3636464206e-3], [23., .9399669854e-3], [24., .2107169269\ e-2], [25., .4204319926e-2], [26., .7605450053e-2], [27., .1264533275e-1], [28. , .1952697749e-1], [29., .2823223424e-1], [30., .3845984162e-1], [31., .\ 4961110752e-1], [32., .6083328955e-1], [33., .7111757138e-1], [34., .7943595055\ e-1], [35., .8489256353e-1], [36., .8686188986e-1], [37., .8508919823e-1], [38. , .7973608995e-1], [39., .7136333858e-1], [40., .6085215031e-1], [41., .\ 4927348190e-1], [42., .3772462671e-1], [43., .2716243137e-1], [44., .1826865334\ e-1], [45., .1137892246e-1], [46., .6490281685e-2], [47., .3338455107e-2], [48. , .1515116499e-2], [49., .5867924716e-3], [50., .1834490527e-3], [51., .\ 4159842464e-4], [52., .5199803080e-5]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [35.93003436, 4.424992848, -.3417357286e-1, 2.743716291] Here is a plot + HHHHH + H HH 0.08 HH HH + H H + HH H + H H 0.06 H H + HH HH + H H + H H + H H 0.04 H H + HH H + H HH + HH HH 0.02 H H + HH HH + HH H + HH HH + HHHH HHH +**************--+-+-+-+-+-+-+--+-+-+-+-+-+-+--+-+-+-+-+-+-+--+-+***********- 0 20 25 30 35 40 45 50 --------------------------------------------- The occupants of cell, [1, 19], in a standard Young tableau of shape, [n, n, n], are all the integers from, 19, to , 55 as n goes to infinity, the distribution is [[19., .1806822354e-6], [20., .2059777484e-5], [21., .1232080767e-4], [22., .\ 5142082581e-4], [23., .1680324714e-3], [24., .4575399146e-3], [25., .1079043233\ e-2], [26., .2262429093e-2], [27., .4296887380e-2], [28., .7495692820e-2], [29. , .1213852570e-1], [30., .1840013733e-1], [31., .2628025029e-1], [32., .\ 3555207271e-1], [33., .4574468669e-1], [34., .5616825567e-1], [35., .6598211488\ e-1], [36., .7429663448e-1], [37., .8029250323e-1], [38., .8333736180e-1], [39. , .8308014772e-1], [40., .7950760522e-1], [41., .7295365825e-1], [42., .\ 6405902984e-1], [43., .5368494790e-1], [44., .4279167632e-1], [45., .3230040786\ e-1], [46., .2296394350e-1], [47., .1527347716e-1], [48., .9422108862e-2], [49. , .5330739371e-2], [50., .2723952421e-2], [51., .1229890265e-2], [52., .\ 4745504495e-3], [53., .1480028161e-3], [54., .3352200535e-4], [55., .4190250669\ e-5]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [38.29330077, 4.599661213, -.4497062499e-1, 2.750006863] Here is a plot + HHHHH 0.08 HH HHH + HH HH + H H + H HH + H HH 0.06 H H + HH H + H H + H H 0.04 H H + HH HH + H H + HH H + HH H 0.02 H HH + HH H + HH HH + HH HHH + HHH HHH +****************-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+***********- 0 20 25 30 35 40 45 50 55 --------------------------------------------- The occupants of cell, [1, 20], in a standard Young tableau of shape, [n, n, n], are all the integers from, 20, to , 58 as n goes to infinity, the distribution is [[20., .6625015299e-7], [21., .7992081948e-6], [22., .5047726303e-5], [23., .\ 2220116238e-4], [24., .7632652473e-4], [25., .2183312155e-3], [26., .5402263112\ e-3], [27., .1187104420e-2], [28., .2360751954e-2], [29., .4308964073e-2], [30. , .7297113648e-2], [31., .1156295434e-1], [32., .1726082246e-1], [33., .\ 2440575885e-1], [34., .3283017450e-1], [35., .4216473662e-1], [36., .5185123017\ e-1], [37., .6118914402e-1], [38., .6941105663e-1], [39., .7577608441e-1], [40. , .7966695771e-1], [41., .8067540021e-1], [42., .7866239018e-1], [43., .\ 7378382357e-1], [44., .6647693068e-1], [45., .5740772057e-1], [46., .4738482403\ e-1], [47., .3725086991e-1], [48., .2776853017e-1], [49., .1952243111e-1], [50. , .1285719730e-1], [51., .7864291647e-2], [52., .4417572854e-2], [53., .\ 2244167242e-2], [54., .1008653197e-2], [55., .3878974905e-3], [56., .1207191264\ e-3], [57., .2731422658e-4], [58., .3414278323e-5]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [40.67183024, 4.770279716, -.5487369516e-1, 2.756039622] Here is a plot 0.08 HHHHH + HH HH + H H + HH HH + H H 0.06 H H + H H + H HH + HH H + H H 0.04 H HH + H H + H HH + HH H + H H 0.02 HH H + HH H + H HH + HHH HHH + HHH HHHH +****************+-+-+-+-++-+-+-+-+-+-+-+-+-+-+-+-++-+-+-+-+-+-+-+**********- 0 20 25 30 35 40 45 50 55 --------------------------------------------- The occupants of cell, [1, 21], in a standard Young tableau of shape, [n, n, n], are all the integers from, 21, to , 61 as n goes to infinity, the distribution is [[21., .2418656379e-7], [22., .3078289937e-6], [23., .2047094674e-5], [24., .\ 9463217032e-5], [25., .3414100423e-4], [26., .1023414679e-3], [27., .2650442020\ e-3], [28., .6089476410e-3], [29., .1265022616e-2], [30., .2410188672e-2], [31. , .4257916432e-2], [32., .7035343456e-2], [33., .1094758116e-1], [34., .\ 1613354211e-1], [35., .2262081140e-1], [36., .3028885380e-1], [37., .3884957306\ e-1], [38., .4785184014e-1], [39., .5671254969e-1], [40., .6477192419e-1], [41. , .7136620531e-1], [42., .7590750264e-1], [43., .7795905677e-1], [44., .\ 7729471836e-1], [45., .7393373866e-1], [46., .6814530954e-1], [47., .6042099423\ e-1], [48., .5141703229e-1], [49., .4187274921e-1], [50., .3251603558e-1], [51. , .2397115617e-1], [52., .1668600104e-1], [53., .1089329127e-1], [54., .\ 6612759744e-2], [55., .3690903394e-2], [56., .1865260155e-2], [57., .8349462444\ e-3], [58., .3201440194e-3], [59., .9944270370e-4], [60., .2247938513e-4], [61. , .2809923141e-5]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [43.06457475, 4.937116817, -.6399761483e-1, 2.761821661] Here is a plot + HHHHH + H HH | HH H + HH H + H HH 0.06 HH H + H H + H H + HH HH + H H 0.04 H HH + H H + H HH + H H + HH HH 0.02 H H + H HH + HH HH + HH HH + HHH HHH +*****************+---+--+---+---+--+---+--+---+---+--+---+---+--***********- 0 30 40 50 60 --------------------------------------------- The occupants of cell, [1, 22], in a standard Young tableau of shape, [n, n, n], are all the integers from, 22, to , 64 as n goes to infinity, the distribution is [[22., .8795114105e-8], [23., .1177780498e-6], [24., .8225874776e-6], [25., .\ 3987118394e-5], [26., .1506039366e-4], [27., .4720466307e-4], [28., .1276800358\ e-3], [29., .3060658164e-3], [30., .6627956445e-3], [31., .1315378342e-2], [32. , .2419030518e-2], [33., .4158672916e-2], [34., .6730524589e-2], [35., .\ 1031371709e-1], [36., .1503497245e-1], [37., .2093177460e-1], [38., .2792094417\ e-1], [39., .3577962584e-1], [40., .4414427161e-1], [41., .5253047694e-1], [42. , .6037303286e-1], [43., .6708199713e-1], [44., .7210766761e-1], [45., .\ 7500561170e-1], [46., .7549264951e-1], [47., .7348584485e-1], [48., .6911874558\ e-1], [49., .6273182616e-1], [50., .5483700083e-1], [51., .4605924343e-1], [52. , .3706193773e-1], [53., .2846632281e-1], [54., .2077820143e-1], [55., .\ 1433534204e-1], [56., .9285569456e-2], [57., .5598685958e-2], [58., .3107061976\ e-2], [59., .1562878235e-2], [60., .6970386416e-3], [61., .2665539739e-3], [62. , .8265394473e-4], [63., .1866865054e-4], [64., .2333581317e-5]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [45.47060102, 5.100412640, -.7243773886e-1, 2.767362472] Here is a plot + HHHHH 0.07 HH H + HH H + H H 0.06 H H + HH H + H H 0.05 H H + H H 0.04 H HH + H H + HH H 0.03 H H + H H + H H 0.02 H HH + HH HH 0.01 HH HH + HH H + HHH HHHH +******************+--+---+--+---+--+---+--+---+--+---+--+---+--+-**********- 0 30 40 50 60 --------------------------------------------- The occupants of cell, [1, 23], in a standard Young tableau of shape, [n, n, n], are all the integers from, 23, to , 67 as n goes to infinity, the distribution is [[23., .3186635545e-8], [24., .4478993294e-7], [25., .3277844136e-6], [26., .\ 1662243519e-5], [27., .6560053176e-5], [28., .2145650230e-4], [29., .6049563997\ e-4], [30., .1510149435e-3], [31., .3402638379e-3], [32., .7020902947e-3], [33. , .1341564684e-2], [34., .2395086115e-2], [35., .4023703770e-2], [36., .\ 6398308851e-2], [37., .9676921915e-2], [38., .1397621211e-1], [39., .1934119287\ e-1], [40., .2571827327e-1], [41., .3293714050e-1], [42., .4070614692e-1], [43. , .4862406272e-1], [44., .5620854789e-1], [45., .6293896215e-1], [46., .\ 6830868208e-1], [47., .7188037788e-1], [48., .7333701119e-1], [49., .7252171678\ e-1], [50., .6946105276e-1], [51., .6436801284e-1], [52., .5762338876e-1], [53. , .4973646174e-1], [54., .4128869509e-1], [55., .3286709262e-1], [56., .\ 2499669744e-1], [57., .1808329476e-1], [58., .1237658751e-1], [59., .7960396321\ e-2], [60., .4770443554e-2], [61., .2633794706e-2], [62., .1319242577e-2], [63. , .5864391311e-3], [64., .2237194263e-3], [65., .6926339910e-4], [66., .\ 1563236438e-4], [67., .1954045548e-5]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [47.88907366, 5.260382968, -.8027397802e-1, 2.772672818] Here is a plot + HHHHH 0.07 H HHH + H HH + HH H 0.06 H HH + H H 0.05 H H + HH H + H HH 0.04 H H + H HH + H H 0.03 HH HH + H H + H H 0.02 HH H + HH HH 0.01 HH H + HH HH + HHHH HHH +*******************+--+--+---+--+--+---+--+--+---+--+--+---+--+-***********- 0 30 40 50 60 --------------------------------------------- The occupants of cell, [1, 24], in a standard Young tableau of shape, [n, n, n], are all the integers from, 24, to , 70 as n goes to infinity, the distribution is [[24., .1150729502e-8], [25., .1693873828e-7], [26., .1296193514e-6], [27., .\ 6863462188e-6], [28., .2824672695e-5], [29., .9623453657e-5], [30., .2823295343\ e-4], [31., .7326690764e-4], [32., .1714742359e-3], [33., .3672422320e-3], [34. , .7278969278e-3], [35., .1347224368e-2], [36., .2345345910e-2], [37., .\ 3863224066e-2], [38., .6050741553e-2], [39., .9048433555e-2], [40., .1296436542\ e-1], [41., .1784903370e-1], [42., .2367219496e-1], [43., .3030591433e-1], [44. , .3751772174e-1], [45., .4497657930e-1], [46., .5227257971e-1], [47., .\ 5894921560e-1], [48., .6454505274e-1], [49., .6864005608e-1], [50., .7090092512\ e-1], [51., .7111971904e-1], [52., .6924074816e-1], [53., .6537198177e-1], [54. , .5977883729e-1], [55., .5286002817e-1], [56., .4510717713e-1], [57., .\ 3705220773e-1], [58., .2920895548e-1], [59., .2201742375e-1], [60., .1579976810\ e-1], [61., .1073565638e-1], [62., .6860994179e-2], [63., .4088921476e-2], [64. , .2246995305e-2], [65., .1121201273e-2], [66., .4969137828e-3], [67., .\ 1891516090e-3], [68., .5847806975e-4], [69., .1318908373e-4], [70., .1648635466\ e-5]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [50.31924138, 5.417222569, -.8757385891e-1, 2.777764018] Here is a plot 0.07 HHHHH + HH HH + HH H 0.06 HH H + H H + HH H 0.05 H H + H H + H H 0.04 HH H + H H + H H 0.03 H HH + HH H 0.02 HH HH + H H + HH H 0.01 HH HH + H HH + HHHH HHH +*******************-+--+--+--+--+--+---+--+--+--+--+--+---+--+--+**********- 0 30 40 50 60 70 --------------------------------------------- The occupants of cell, [1, 25], in a standard Young tableau of shape, [n, n, n], are all the integers from, 25, to , 73 as n goes to infinity, the distribution is [[25., .4142626209e-9], [26., .6373271091e-8], [27., .5089804448e-7], [28., .\ 2809007431e-6], [29., .1203474222e-5], [30., .4263704197e-5], [31., .1299488644\ e-4], [32., .3500241118e-4], [33., .8496021577e-4], [34., .1885756786e-3], [35. , .3871191044e-3], [36., .7416796390e-3], [37., .1335916703e-2], [38., .\ 2275868047e-2], [39., .3685479449e-2], [40., .5697002601e-2], [41., .8436239355\ e-2], [42., .1200366475e-1], [43., .1645353270e-1], [44., .2177391927e-1], [45. , .2787107562e-1], [46., .3456130830e-1], [47., .4157285312e-1], [48., .\ 4855895272e-1], [49., .5512178417e-1], [50., .6084526710e-1], [51., .6533338671\ e-1], [52., .6824971652e-1], [53., .6935346851e-1], [54., .6852765814e-1], [55. , .6579575706e-1], [56., .6132435230e-1], [57., .5541069378e-1], [58., .\ 4845555092e-1], [59., .4092355356e-1], [60., .3329514919e-1], [61., .2601620963\ e-1], [62., .1945260820e-1], [63., .1385707247e-1], [64., .9353843063e-2], [65. , .5943264174e-2], [66., .3524208326e-2], [67., .1928446950e-2], [68., .\ 9589060660e-3], [69., .4238261328e-3], [70., .1610082759e-3], [71., .4971270020\ e-4], [72., .1120508481e-4], [73., .1400635601e-5]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [52.76042555, 5.571107938, -.9439487583e-1, 2.782647479] Here is a plot 0.07 HHHH + HH HHH + H HH 0.06 H HH + H H + H H 0.05 H H + HH HH 0.04 H H + H H + H HH 0.03 HH H + H H + H HH 0.02 H HH + HH HH + H HH 0.01 H HH + HHH H + HHHH HHH +********************+--+--+--+--+--+---+--+--+--+--+--+--+--+--+-**********- 0 30 40 50 60 70 --------------------------------------------- The occupants of cell, [1, 26], in a standard Young tableau of shape, [n, n, n], are all the integers from, 26, to , 76 as n goes to infinity, the distribution is [[26., .1487096588e-9], [27., .2386698227e-8], [28., .1985746039e-7], [29., .\ 1140327695e-6], [30., .5077860952e-6], [31., .1867901294e-5], [32., .5905506926\ e-5], [33., .1648664267e-4], [34., .4144438837e-4], [35., .9520303708e-4], [36. , .2021414396e-3], [37., .4003445375e-3], [38., .7450662041e-3], [39., .\ 1310935650e-2], [40., .2191784676e-2], [41., .3497044621e-2], [42., .5344010155\ e-2], [43., .7845879431e-2], [44., .1109627381e-1], [45., .1515177464e-1], [46. , .2001471860e-1], [47., .2561890391e-1], [48., .3182085933e-1], [49., .\ 3839887195e-1], [50., .4506109962e-1], [51., .5146292986e-1], [52., .5723246759\ e-1], [53., .6200183989e-1], [54., .6544108676e-1], [55., .6729090081e-1], [56. , .6739044999e-1], [57., .6569693839e-1], [58., .6229433127e-1], [59., .\ 5738967522e-1], [60., .5129661083e-1], [61., .4440701018e-1], [62., .3715319518\ e-1], [63., .2996482066e-1], [64., .2322593204e-1], [65., .1723843461e-1], [66. , .1219773458e-1], [67., .8184409574e-2], [68., .5172707680e-2], [69., .\ 3053221722e-2], [70., .1664240814e-2], [71., .8248990006e-3], [72., .3636864064\ e-3], [73., .1379087361e-3], [74., .4252984112e-4], [75., .9580542596e-5], [76. , .1197567825e-5]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [55.21201077, 5.722199621, -.1007863205, 2.787334410] Here is a plot + HHHHH + H HH 0.06 HH H + HH H + HH HH 0.05 HH HH + H H + HH H 0.04 H HH + H H + H H 0.03 HH H + HH H + H H 0.02 H HH + H HH + HH HH 0.01 HH HH + HH HH + HHHH HHH +*********************+--+--+--+--+--+-+--+--+--+--+--+--+--+--+--**********- 0 30 40 50 60 70 --------------------------------------------- The occupants of cell, [1, 27], in a standard Young tableau of shape, [n, n, n], are all the integers from, 27, to , 79 as n goes to infinity, the distribution is [[27., .5324172969e-10], [28., .8898974819e-9], [29., .7701079614e-8], [30., .\ 4594564080e-7], [31., .2123366099e-6], [32., .8098567576e-6], [33., .2652392087\ e-5], [34., .7664572818e-5], [35., .1992858760e-4], [36., .4731794064e-4], [37. , .1037843510e-3], [38., .2122150803e-3], [39., .4075620770e-3], [40., .\ 7396966510e-3], [41., .1275223597e-2], [42., .2097368982e-2], [43., .3303102204\ e-2], [44., .4996904667e-2], [45., .7281050699e-2], [46., .1024286799e-1], [47. , .1394009783e-1], [48., .1838605669e-1], [49., .2353668590e-1], [50., .\ 2928166644e-1], [51., .3544152117e-1], [52., .4177203628e-1], [53., .4797647863\ e-1], [54., .5372508634e-1], [55., .5868031275e-1], [56., .6252546123e-1], [57. , .6499378287e-1], [58., .6589490001e-1], [59., .6513557749e-1], [60., .\ 6273234249e-1], [61., .5881416780e-1], [62., .5361431066e-1], [63., .4745141532\ e-1], [64., .4070115947e-1], [65., .3376103914e-1], [66., .2701220628e-1], [67. , .2078328705e-1], [68., .1532140248e-1], [69., .1077484377e-1], [70., .\ 7189961212e-2], [71., .4522146022e-2], [72., .2657996688e-2], [73., .1443654553\ e-2], [74., .7134729946e-3], [75., .3138388061e-3], [76., .1188063318e-3], [77. , .3659868653e-4], [78., .8240055155e-5], [79., .1030006894e-5]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [57.67343690, 5.870644151, -.1067907203, 2.791835638] Here is a plot + HHHHH + HH HH 0.06 H H + H H + H H 0.05 HH H + H HH + H H 0.04 H HH + HH H + HH H 0.03 H H + H HH + H H 0.02 HH H + H HH + HH H 0.01 HHH HH + HH HH + HHHH HHH +**********************+-+--+--+--+--+-+--+--+--+--+-+--+--+--+--+**********+ 0 30 40 50 60 70 80 --------------------------------------------- The occupants of cell, [1, 28], in a standard Young tableau of shape, [n, n, n], are all the integers from, 28, to , 82 as n goes to infinity, the distribution is [[28., .1901490346e-10], [29., .3304659084e-9], [30., .2970127920e-8], [31., .\ 1838389037e-7], [32., .8805568659e-7], [33., .3477636800e-6], [34., .1178400717\ e-5], [35., .3520366391e-5], [36., .9456147928e-5], [37., .2318039986e-4], [38. , .5246008273e-4], [39., .1106226858e-3], [40., .2189903238e-3], [41., .\ 4095089502e-3], [42., .7271313710e-3], [43., .1231343567e-2], [44., .1996128513\ e-2], [45., .3107689168e-2], [46., .4659433683e-2], [47., .6744062737e-2], [48. , .9443057284e-2], [49., .1281437829e-1], [50., .1687966896e-1], [51., .\ 2161260026e-1], [52., .2693014028e-1], [53., .3268840844e-1], [54., .3868438655\ e-1], [55., .4466414043e-1], [56., .5033743539e-1], [57., .5539781250e-1], [58. , .5954644793e-1], [59., .6251754998e-1], [60., .6410273688e-1], [61., .\ 6417181815e-1], [62., .6268765824e-1], [63., .5971328297e-1], [64., .5541003430\ e-1], [65., .5002635067e-1], [66., .4387764750e-1], [67., .3731880448e-1], [68. , .3071188362e-1], [69., .2439274045e-1], [70., .1864085854e-1], [71., .\ 1365671009e-1], [72., .9550009349e-2], [73., .6340426060e-2], [74., .3970021045\ e-2], [75., .2324437852e-2], [76., .1258347505e-2], [77., .6202182132e-3], [78. , .2722414774e-3], [79., .1028992109e-3], [80., .3166636959e-4], [81., .\ 7126046604e-5], [82., .8907558254e-6]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [60.14419234, 6.016575704, -.1124449817, 2.796161512] Here is a plot + HHHHH 0.06 HH HH + HH HH + HH HH 0.05 HH HH + HH H + H HH + HH H 0.04 H H + H H + H HH 0.03 H H + HH HH + H H 0.02 H H + H H + H H 0.01 HH H + HHH HH + HHHH HHHH +**********************+--+--+-+--+--+-+--+--+-+--+--+--+-+--+--+-**********- 0 30 40 50 60 70 80 --------------------------------------------- The occupants of cell, [1, 29], in a standard Young tableau of shape, [n, n, n], are all the integers from, 29, to , 85 as n goes to infinity, the distribution is [[29., .6775425371e-11], [30., .1222587867e-9], [31., .1139631404e-8], [32., .\ 7308425500e-8], [33., .3623562175e-7], [34., .1480062507e-6], [35., .5182754216\ e-6], [36., .1598851979e-5], [37., .4431936708e-5], [38., .1120441032e-4], [39. , .2613595520e-4], [40., .5677657626e-4], [41., .1157339650e-3], [42., .\ 2227542772e-3], [43., .4069474719e-3], [44., .7088000349e-3], [45., .1181485527\ e-2], [46., .1890905611e-2], [47., .2913907828e-2], [48., .4334256415e-2], [49. , .6236183256e-2], [50., .8695696760e-2], [51., .1177023099e-1], [52., .\ 1548761145e-1], [53., .1983562556e-1], [54., .2475364917e-1], [55., .3012774890\ e-1], [56., .3579043655e-1], [57., .4152581244e-1], [58., .4708024920e-1], [59. , .5217811057e-1], [60., .5654135945e-1], [61., .5991137343e-1], [62., .\ 6207092768e-1], [63., .6286416763e-1], [64., .6221248710e-1], [65., .6012452536\ e-1], [66., .5669895451e-1], [67., .5211930762e-1], [68., .4664078426e-1], [69. , .4056976888e-1], [70., .3423770128e-1], [71., .2797187162e-1], [72., .\ 2206649934e-1], [73., .1675784323e-1], [74., .1220683481e-1], [75., .8491715006\ e-2], [76., .5611496828e-2], [77., .3499120249e-2], [78., .2041401230e-2], [79. , .1101776697e-2], [80., .5416962616e-3], [81., .2373096382e-3], [82., .\ 8956703403e-4], [83., .2753766515e-4], [84., .6194108963e-5], [85., .7742636204\ e-6]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [62.62380835, 6.160117507, -.1177813060, 2.800321842] Here is a plot + HHHH 0.06 HH HH + H HH + H HH 0.05 H H + H H + HH H 0.04 H H + H HH + HH H 0.03 HH HH + H H + HH H + HH H 0.02 HH H + HH HH + HH H 0.01 HH H + HH HH + HHH HHHH +***********************+-+--+-+--+--+-+--+--+-+--+-+--+--+-+--+-+**********+ 0 30 40 50 60 70 80 --------------------------------------------- The occupants of cell, [1, 30], in a standard Young tableau of shape, [n, n, n], are all the integers from, 30, to , 88 as n goes to infinity, the distribution is [[30., .2409040132e-11], [31., .4507236376e-10], [32., .4351830621e-9], [33., .\ 2887992999e-8], [34., .1480438628e-7], [35., .6246873427e-7], [36., .2258097858\ e-6], [37., .7185974796e-6], [38., .2053453371e-5], [39., .5348528207e-5], [40. , .1284689802e-4], [41., .2872261499e-4], [42., .6022956131e-4], [43., .\ 1192028540e-3], [44., .2238444863e-3], [45., .4006201627e-3], [46., .6859775828\ e-3], [47., .1127491321e-2], [48., .1783974738e-2], [49., .2724103078e-2], [50. , .4023184291e-2], [51., .5757900995e-2], [52., .7999115971e-2], [53., .\ 1080315309e-1], [54., .1420228926e-1], [55., .1819546605e-1], [56., .2274039946\ e-1], [57., .2774829130e-1], [58., .3308220452e-1], [59., .3855986048e-1], [60. , .4396117878e-1], [61., .4904035880e-1], [62., .5354176446e-1], [63., .\ 5721838788e-1], [64., .5985129858e-1], [65., .6126827869e-1], [66., .6135982282\ e-1], [67., .6009083802e-1], [68., .5750668593e-1], [69., .5373263462e-1], [70. , .4896630826e-1], [71., .4346333808e-1], [72., .3751713090e-1], [73., .\ 3143445469e-1], [74., .2550930281e-1], [75., .1999806822e-1], [76., .1509922650\ e-1], [77., .1094031854e-1], [78., .7573999306e-2], [79., .4983425719e-2], [80. , .3095613946e-2], [81., .1800012421e-2], [82., .9687685972e-3], [83., .\ 4752034695e-3], [84., .2078023652e-3], [85., .7832550689e-4], [86., .2406045014\ e-4], [87., .5409669835e-5], [88., .6762087294e-6]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [65.11185419, 6.301383040, -.1228279294, 2.804325882] Here is a plot 0.06 HHHHH + HH H + HH HH + HH H 0.05 H H + H HH + H HH 0.04 HH H + H HH + H H 0.03 H H + H H + H H + H H 0.02 H HH + H H + H HH 0.01 HH H + HHH HH + HHHH HHH +************************--+-+--+-+--+-+--+-+--+-+--+-+--+-+--+-+-**********- 0 30 40 50 60 70 80 --------------------------------------------- The occupants of cell, [1, 31], in a standard Young tableau of shape, [n, n, n], are all the integers from, 31, to , 91 as n goes to infinity, the distribution is [[31., .8548206919e-12], [32., .1656215091e-10], [33., .1654380248e-9], [34., .\ 1134821959e-8], [35., .6007986529e-8], [36., .2616208057e-7], [37., .9752389838\ e-7], [38., .3198324107e-6], [39., .9412838924e-6], [40., .2523592287e-5], [41. , .6235927479e-5], [42., .1433610659e-4], [43., .3089762939e-4], [44., .\ 6282474242e-4], [45., .1211590072e-3], [46., .2226178487e-3], [47., .3912221927\ e-3], [48., .6597775700e-3], [49., .1070888551e-2], [50., .1677131747e-2], [51. , .2540009459e-2], [52., .3727370838e-2], [53., .5309126528e-2], [54., .\ 7351289319e-2], [55., .9908626086e-2], [56., .1301647154e-1], [57., .1668249107\ e-1], [58., .2087934567e-1], [59., .2553927153e-1], [60., .3055151694e-1], [61. , .3576337619e-1], [62., .4098523829e-1], [63., .4599966195e-1], [64., .\ 5057404434e-1], [65., .5447602426e-1], [66., .5749040450e-1], [67., .5943613748\ e-1], [68., .6018181990e-1], [69., .5965819363e-1], [70., .5786633765e-1], [71. , .5488053548e-1], [72., .5084519320e-1], [73., .4596565408e-1], [74., .\ 4049331099e-1], [75., .3470605050e-1], [76., .2888573299e-1], [77., .2329501889\ e-1], [78., .1815623863e-1], [79., .1363500255e-1], [80., .9830744608e-2], [81. , .6775399622e-2], [82., .4440088866e-2], [83., .2748323170e-2], [84., .\ 1593155965e-2], [85., .8552019649e-3], [86., .4185970458e-3], [87., .1827399558\ e-3], [88., .6879333163e-4], [89., .2111516291e-4], [90., .4745579542e-5], [91. , .5931974428e-6]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [67.60793291, 6.440477105, -.1276097269, 2.808182327] Here is a plot 0.06 HHHH + HHH HH + H HH 0.05 H H + H H + H H + HH H 0.04 H H + H H + HH H 0.03 HH H + H H + HH H 0.02 HH H + HH H + HH HH + HH HH 0.01 HH HH + HH HH + HHHH HH +************************+-+-+--+-+--+-+--+-+--+-+-+--+-+--+-+--+-**********+ 0 40 50 60 70 80 90 --------------------------------------------- The occupants of cell, [1, 32], in a standard Young tableau of shape, [n, n, n], are all the integers from, 32, to , 94 as n goes to infinity, the distribution is [[32., .3027489951e-12], [33., .6067212184e-11], [34., .6262946695e-10], [35., .4435833424e-9], [36., .2422917876e-8], [37., .1087741054e-7], [38., .\ 4177443488e-7], [39., .1410558940e-6], [40., .4271697867e-6], [41., .1177795889\ e-5], [42., .2991573389e-5], [43., .7065951664e-5], [44., .1563918534e-4], [45. , .3264322077e-4], [46., .6459959636e-4], [47., .1217583627e-3], [48., .\ 2194290695e-3], [49., .3793860858e-3], [50., .6311561786e-3], [51., .1012927571\ e-2], [52., .1571773085e-2], [53., .2362872227e-2], [54., .3447461575e-2], [55. , .4889346137e-2], [56., .6749962978e-2], [57., .9082188413e-2], [58., .\ 1192329714e-1], [59., .1528768485e-1], [60., .1916012192e-1], [61., .2349038403\ e-1], [62., .2819008508e-1], [63., .3313240895e-1], [64., .3815520549e-1], [65. , .4306760409e-1], [66., .4765993821e-1], [67., .5171640805e-1], [68., .\ 5502957975e-1], [69., .5741556842e-1], [70., .5872860739e-1], [71., .5887368218\ e-1], [72., .5781600240e-1], [73., .5558628417e-1], [74., .5228110018e-1], [75. , .4805791261e-1], [76., .4312483277e-1], [77., .3772565179e-1], [78., .\ 3212124527e-1], [79., .2656901803e-1], [80., .2130252347e-1], [81., .1651362817\ e-1], [82., .1233946748e-1], [83., .8855884514e-2], [84., .6078107058e-2], [85. , .3968256973e-2], [86., .2448159297e-2], [87., .1415089713e-2], [88., .\ 7577707682e-3], [89., .3701661109e-3], [90., .1613429635e-3], [91., .6066792028\ e-4], [92., .1860707585e-4], [93., .4180344268e-5], [94., .5225430335e-6]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [70.11167779, 6.577496695, -.1321487071, 2.811899328] Here is a plot 0.06 HHHHH + H HH + H HH 0.05 H HH + HH HH + HH H 0.04 H H + H H + HH H + H H 0.03 H H + H H + H HH 0.02 H H + H H + H H + H HH 0.01 HH HH + HH HH + HHH HHH +*************************-+--+-+-+--+-+--+-+-+--+-+-+--+-+-+--+-+**********- 0 40 50 60 70 80 90 --------------------------------------------- The occupants of cell, [1, 33], in a standard Young tableau of shape, [n, n, n], are all the integers from, 33, to , 97 as n goes to infinity, the distribution is [[33., .1070324730e-12], [34., .2216201794e-11], [35., .2361646033e-10], [36., .1725363465e-9], [37., .9713805541e-9], [38., .4491783059e-8], [39., .\ 1775674210e-7], [40., .6167920407e-7], [41., .1920417277e-6], [42., .5441038625\ e-6], [43., .1419426346e-5], [44., .3441794133e-5], [45., .7817056547e-5], [46. , .1673650121e-4], [47., .3396135693e-4], [48., .6561326751e-4], [49., .\ 1211690193e-3], [50., .2146164736e-3], [51., .3656748437e-3], [52., .6009223829\ e-3], [53., .9546181718e-3], [54., .1468964605e-2], [55., .2193546489e-2], [56. , .3183712373e-2], [57., .4497740007e-2], [58., .6192749597e-2], [59., .\ 8319486725e-2], [60., .1091627405e-1], [61., .1400260412e-1], [62., .1757298834\ e-1], [63., .2159176461e-1], [64., .2598957779e-1], [65., .3066217278e-1], [66. , .3547197878e-1], [67., .4025272800e-1], [68., .4481706470e-1], [69., .\ 4896679288e-1], [70., .5250511670e-1], [71., .5524998236e-1], [72., .5704746045\ e-1], [73., .5778403333e-1], [74., .5739667697e-1], [75., .5587974579e-1], [76. , .5328786802e-1], [77., .4973432660e-1], [78., .4538472918e-1], [79., .\ 4044616180e-1], [80., .3515247172e-1], [81., .2974681147e-1], [82., .2446304223\ e-1], [83., .1950794177e-1], [84., .1504627784e-1], [85., .1119059484e-1], [86. , .7996990199e-2], [87., .5467289451e-2], [88., .3557024140e-2], [89., .\ 2187692289e-2], [90., .1261151317e-2], [91., .6738053908e-3], [92., .3285351125\ e-3], [93., .1429867755e-3], [94., .5370736420e-4], [95., .1646057128e-4], [96. , .3696818609e-5], [97., .4621023262e-6]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [72.62274920, 6.712531813, -.1364644213, 2.815484510] Here is a plot + HHHH + HH HH + HH H 0.05 H H + HH HH + H HH 0.04 HH H + H HH + H H + HH H 0.03 HH HH + H H + H HH 0.02 HH HH + HH H + HH H + HH H 0.01 HH HH + HHH HH + HHH HHH +*************************--+-+-+--+-+-+-+--+-+-+--+-+-+-+--+-+-+-**********+ 0 40 50 60 70 80 90 --------------------------------------------- The occupants of cell, [1, 34], in a standard Young tableau of shape, [n, n, n], are all the integers from, 34, to , 100 as n goes to infinity, the distribution is [[34., .3777616694e-13], [35., .8073306535e-12], [36., .8872502321e-11], [37., .6679945609e-10], [38., .3872892185e-9], [39., .1843021821e-8], [40., .\ 7493260108e-8], [41., .2675402738e-7], [42., .8557598783e-7], [43., .2489557400\ e-6], [44., .6665452578e-6], [45., .1657998763e-5], [46., .3861395384e-5], [47. , .8474209315e-5], [48., .1761964059e-4], [49., .3486880630e-4], [50., .\ 6593833599e-4], [51., .1195609910e-3], [52., .2084933070e-3], [53., .3505806638\ e-3], [54., .5697512894e-3], [55., .8967640831e-3], [56., .1369500465e-2], [57. , .2032578065e-2], [58., .2936082967e-2], [59., .4133273136e-2], [60., .\ 5677198909e-2], [61., .7616311454e-2], [62., .9989274528e-2], [63., .1281934153\ e-1], [64., .1610878813e-1], [65., .1983398010e-1], [66., .2394168855e-1], [67. , .2834722906e-1], [68., .3293489335e-1], [69., .3756096851e-1], [70., .\ 4205941338e-1], [71., .4625000740e-1], [72., .4994853123e-1], [73., .5297830826\ e-1], [74., .5518225848e-1], [75., .5643450945e-1], [76., .5665058395e-1], [77. , .5579524047e-1], [78., .5388717240e-1], [79., .5099996602e-1], [80., .\ 4725896599e-1], [81., .4283400025e-1], [82., .3792827287e-1], [83., .3276413671\ e-1], [84., .2756687739e-1], [85., .2254801670e-1], [86., .1788988819e-1], [87. , .1373325804e-1], [88., .1016949291e-1], [89., .7238209134e-2], [90., .\ 4930547984e-2], [91., .3197358474e-2], [92., .1960815915e-2], [93., .1127533134\ e-2], [94., .6011366711e-3], [95., .2925908484e-3], [96., .1271674536e-3], [97. , .4771691919e-4], [98., .1461485979e-4], [99., .3281228543e-5], [100., .\ 4101535679e-6]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [75.14083185, 6.845666132, -.1405743036, 2.818945004] Here is a plot + HHHH + H HH 0.05 H H + H H + H H + H HH 0.04 H H + H H + HH HH 0.03 HH H + H HH + HH HH + HH H 0.02 H HH + H HH + H H + HH H 0.01 HH H + HH H + HHHH HHH +**************************-+-+-+--+-+-+-+--+-+-+-+-+--+-+-+-+-+--**********- 0 40 50 60 70 80 90 100 --------------------------------------------- The occupants of cell, [1, 35], in a standard Young tableau of shape, [n, n, n], are all the integers from, 35, to , 103 as n goes to infinity, the distribution is [[35., .1331160168e-13], [36., .2933482593e-12], [37., .3321744305e-11], [38., .2574956788e-10], [39., .1536092642e-9], [40., .7516632176e-9], [41., .\ 3140631080e-8], [42., .1151719347e-7], [43., .3781756679e-7], [44., .1128845900\ e-6], [45., .3099651714e-6], [46., .7904067891e-6], [47., .1886333383e-5], [48. , .4240501171e-5], [49., .9028304153e-5], [50., .1828923580e-4], [51., .\ 3539277873e-4], [52., .6565438042e-4], [53., .1170990914e-3], [54., .2013430155\ e-3], [55., .3345272233e-3], [56., .5381987346e-3], [57., .8399943559e-3], [58. , .1273953046e-2], [59., .1880269263e-2], [60., .2704310818e-2], [61., .\ 3794765279e-2], [62., .5200850170e-2], [63., .6968620802e-2], [64., .9136526993\ e-2], [65., .1173049342e-1], [66., .1475891202e-1], [67., .1820802163e-1], [68. , .2203819526e-1], [69., .2618164658e-1], [70., .3054199965e-1], [71., .\ 3499604127e-1], [72., .3939780121e-1], [73., .4358489988e-1], [74., .4738688357\ e-1], [75., .5063505973e-1], [76., .5317316909e-1], [77., .5486810765e-1], [78. , .5561985238e-1], [79., .5536975297e-1], [80., .5410642754e-1], [81., .\ 5186863375e-1], [82., .4874467184e-1], [83., .4486810831e-1], [84., .4040988835\ e-1], [85., .3556722970e-1], [86., .3055004883e-1], [87., .2556602730e-1], [88. , .2080572240e-1], [89., .1642928568e-1], [90., .1255629988e-1], [91., .\ 9259938205e-2], [92., .6566101567e-2], [93., .4457479947e-2], [94., .2881746831\ e-2], [95., .1762486558e-2], [96., .1011108367e-2], [97., .5379916955e-3], [98. , .2614268619e-3], [99., .1134758205e-3], [100., .4253878433e-4], [101., .\ 1302074819e-4], [102., .2922434593e-5], [103., .3653043242e-6]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [77.66563248, 6.976977624, -.1444939556, 2.822287469] Here is a plot + HHH + HH HH 0.05 H HH + H HH + HH H + H H 0.04 H H + HH H + HH H 0.03 H H + HH H + HH HH + H H 0.02 H HH + H H + HH H 0.01 HH H + HH H + HHH HH + HHH HHH +**************************-+-+--+-+-+-+-+-+--+-+-+-+-+-+-+--+-+-+**********+ 0 40 50 60 70 80 90 100 --------------------------------------------- The occupants of cell, [1, 36], in a standard Young tableau of shape, [n, n, n], are all the integers from, 36, to , 106 as n goes to infinity, the distribution is [[36., .4683711704e-14], [37., .1063329144e-12], [38., .1239540194e-11], [39., .9885060286e-11], [40., .6062671912e-10], [41., .3048217463e-9], [42., .\ 1307885164e-8], [43., .4922637463e-8], [44., .1658155718e-7], [45., .5075067395\ e-7], [46., .1428244478e-6], [47., .3731139777e-6], [48., .9118855459e-6], [49. , .2098507883e-5], [50., .4572165776e-5], [51., .9475334738e-5], [52., .\ 1875285892e-4], [53., .3556727019e-4], [54., .6484312930e-4], [55., .1139382619\ e-3], [56., .1934173246e-3], [57., .3178741004e-3], [58., .5067159434e-3], [59. , .7847912877e-3], [60., .1182714917e-2], [61., .1736732262e-2], [62., .\ 2487969445e-2], [63., .3480944694e-2], [64., .4761270834e-2], [65., .6372556124\ e-2], [66., .8352605895e-2], [67., .1072913094e-1], [68., .1351526801e-1], [69. , .1670529971e-1], [70., .2027101234e-1], [71., .2415914023e-1], [72., .\ 2829030723e-1], [73., .3255978942e-1], [74., .3684029208e-1], [75., .4098676919\ e-1], [76., .4484313063e-1], [77., .4825049829e-1], [78., .5105650684e-1], [79. , .5312501474e-1], [80., .5434550998e-1], [81., .5464146948e-1], [82., .\ 5397696330e-1], [83., .5236088013e-1], [84., .4984828337e-1], [85., .4653858259\ e-1], [86., .4257042127e-1], [87., .3811343954e-1], [88., .3335736487e-1], [89. , .2849919850e-1], [90., .2372956430e-1], [91., .1921951314e-1], [92., .\ 1510916437e-1], [93., .1149945941e-1], [94., .8447977143e-2], [95., .5969242248\ e-2], [96., .4039325901e-2], [97., .2603912639e-2], [98., .1588518208e-2], [99. , .9092958557e-3], [100., .4829133328e-3], [101., .2343007389e-3], [102., .\ 1015776393e-3], [103., .3804430619e-4], [104., .1163817890e-4], [105., .\ 2611371318e-5], [106., .3264214148e-6]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [80.19687767, 7.106539064, -.1482373852, 2.825518128] Here is a plot + HHHH + HH HH 0.05 HH H + H H + H HH 0.04 H H + H H + H H + H H 0.03 HH H + H H + H H + HH HH 0.02 HH H + HH HH + H HH 0.01 H H + HH HH + HH HH + HHH HHH +***************************-+-+-+-+-+-+-+-+-+-+-+--+-+-+-+-+-+-+-+*********- 0 40 50 60 70 80 90 100 --------------------------------------------- The occupants of cell, [1, 37], in a standard Young tableau of shape, [n, n, n], are all the integers from, 37, to , 109 as n goes to infinity, the distribution is [[37., .1645628436e-14], [38., .3845573820e-13], [39., .4611135270e-12], [40., .3780069468e-11], [41., .2381734396e-10], [42., .1229517560e-9], [43., .\ 5413558477e-9], [44., .2089843351e-8], [45., .7216633674e-8], [46., .2263327123\ e-7], [47., .6524064258e-7], [48., .1744986689e-6], [49., .4364765460e-6], [50. , .1027652094e-5], [51., .2289954657e-5], [52., .4852131928e-5], [53., .\ 9815475746e-5], [54., .1902311883e-4], [55., .3543011790e-4], [56., .6358494911\ e-4], [57., .1102207489e-3], [58., .1849362294e-3], [59., .3009223498e-3], [60. , .4756635338e-3], [61., .7315148769e-3], [62., .1096033918e-2], [63., .\ 1601932340e-2], [64., .2286514437e-2], [65., .3190489329e-2], [66., .4356084734\ e-2], [67., .5824450862e-2], [68., .7632419996e-2], [69., .9808773546e-2], [70. , .1237025463e-1], [71., .1531763956e-1], [72., .1863223509e-1], [73., .\ 2227319046e-1], [74., .2617599700e-1], [75., .3025249096e-1], [76., .3439257874\ e-1], [77., .3846777486e-1], [78., .4233649185e-1], [79., .4585086262e-1], [80. , .4886472454e-1], [81., .5124226560e-1], [82., .5286674006e-1], [83., .\ 5364861186e-1], [84., .5353248358e-1], [85., .5250221542e-1], [86., .5058372898\ e-1], [87., .4784511997e-1], [88., .4439386944e-1], [89., .4037114489e-1], [90. , .3594341936e-1], [91., .3129190213e-1], [92., .2660054841e-1], [93., .\ 2204366111e-1], [94., .1777426401e-1], [95., .1391445675e-1], [96., .1054881744\ e-1], [97., .7721587632e-2], [98., .5437891677e-2], [99., .3668684056e-2], [100\ ., .2358590711e-2], [101., .1435420632e-2], [102., .8199542409e-3], [103., .\ 4346976615e-3], [104., .2106010677e-3], [105., .9119833581e-4], [106., .\ 3412798576e-4], [107., .1043434144e-4], [108., .2340617472e-5], [109., .\ 2925771840e-6]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [82.73431211, 7.234418535, -.1518172095, 2.828642789] Here is a plot + HHHH 0.05 HH H + H H + H H + HH H 0.04 HH HH + H H + H HH + HH H 0.03 H H + H H + HH H + HH H 0.02 H H + H H + H HH 0.01 H H + H H + HH HH + HHH HHH +***************************-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+**********+ 0 40 50 60 70 80 90 100 110 --------------------------------------------- The occupants of cell, [1, 38], in a standard Young tableau of shape, [n, n, n], are all the integers from, 38, to , 112 as n goes to infinity, the distribution is [[38., .5774134865e-15], [39., .1387766431e-13], [40., .1710330825e-12], [41., .1440197549e-11], [42., .9315659266e-11], [43., .4934179380e-10], [44., .\ 2227912071e-9], [45., .8815590445e-9], [46., .3118845740e-8], [47., .1001701380\ e-7], [48., .2955707948e-7], [49., .8089438592e-7], [50., .2069713999e-6], [51. , .4982738054e-6], [52., .1134957353e-5], [53., .2457438839e-5], [54., .\ 5078482647e-5], [55., .1005217372e-4], [56., .1911602518e-4], [57., .3502071142\ e-4], [58., .6195641629e-4], [59., .1060746471e-3], [60., .1760892329e-3], [61. , .2839205667e-3], [62., .4453244736e-3], [63., .6804239862e-3], [64., .\ 1014042342e-2], [65., .1475722812e-2], [66., .2099319671e-2], [67., .2922058216\ e-2], [68., .3982992219e-2], [69., .5320834667e-2], [70., .6971199604e-2], [71. , .8963364507e-2], [72., .1131673676e-1], [73., .1403727593e-1], [74., .\ 1711417646e-1], [75., .2051714493e-1], [76., .2419460510e-1], [77., .2807312958\ e-1], [78., .3205832782e-1], [79., .3603732141e-1], [80., .3988281551e-1], [81. , .4345864037e-1], [82., .4662650246e-1], [83., .4925356231e-1], [84., .\ 5122035859e-1], [85., .5242853403e-1], [86., .5280779439e-1], [87., .5232154824\ e-1], [88., .5097073108e-1], [89., .4879540909e-1], [90., .4587388202e-1], [91. , .4231916118e-1], [92., .3827288570e-1], [93., .3389695788e-1], [94., .\ 2936341547e-1], [95., .2484329444e-1], [96., .2049543370e-1], [97., .1645628772\ e-1], [98., .1283179932e-1], [99., .9692214269e-2], [100., .7070392222e-2], [ 101., .4963724406e-2], [102., .3339278304e-2], [103., .2141346706e-2], [104., .\ 1300270912e-2], [105., .7412985984e-3], [106., .3923449830e-3], [107., .\ 1898216947e-3], [108., .8211113520e-4], [109., .3070283648e-4], [110., .\ 9382211939e-5], [111., .2104063839e-5], [112., .2630079799e-6]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [85.27769688, 7.360679795, -.1552448258, 2.831666882] Here is a plot + HHHH 0.05 HH HH + H H + HH H + HH H 0.04 H HH + H H + HH HH + H H 0.03 H H + HH H + H H 0.02 H H + HH H + H H + H H 0.01 H HH + HH HH + HH HH + HHHH HH +****************************+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-*********- 0 40 50 60 70 80 90 100 110 --------------------------------------------- The occupants of cell, [1, 39], in a standard Young tableau of shape, [n, n, n], are all the integers from, 39, to , 115 as n goes to infinity, the distribution is [[39., .2023414782e-15], [40., .4997834511e-14], [41., .6326214540e-13], [42., .5468016153e-12], [43., .3628479939e-11], [44., .1970615671e-10], [45., .\ 9118965164e-10], [46., .3696200759e-9], [47., .1338941629e-8], [48., .\ 4401359702e-8], [49., .1328672502e-7], [50., .3718938045e-7], [51., .9727477126\ e-7], [52., .2393321128e-6], [53., .5569515611e-6], [54., .1231673012e-5], [55. , .2598956802e-5], [56., .5251247697e-5], [57., .1019130791e-4], [58., .\ 1904963624e-4], [59., .3437827644e-4], [60., .6002874216e-4], [61., .1016134213\ e-3], [62., .1670373521e-3], [63., .2670710072e-3], [64., .4159158044e-3], [65. , .6316944928e-3], [66., .9367811092e-3], [67., .1357873188e-2], [68., .\ 1925705744e-2], [69., .2674315222e-2], [70., .3639784133e-2], [71., .4858433894\ e-2], [72., .6364483129e-2], [73., .8187247834e-2], [74., .1034802312e-1], [75. , .1285684693e-1], [76., .1570939686e-1], [77., .1888430442e-1], [78., .\ 2234118103e-1], [79., .2601963214e-1], [80., .2983948914e-1], [81., .3370241415\ e-1], [82., .3749493612e-1], [83., .4109286430e-1], [84., .4436690792e-1], [85. , .4718921805e-1], [86., .4944047155e-1], [87., .5101704419e-1], [88., .\ 5183777989e-1], [89., .5184985615e-1], [90., .5103327429e-1], [91., .4940356397\ e-1], [92., .4701238228e-1], [93., .4394580685e-1], [94., .4032026905e-1], [95. , .3627624812e-1], [96., .3197004487e-1], [97., .2756416439e-1], [98., .\ 2321703706e-1], [99., .1907296269e-1], [100., .1525323393e-1], [101., .\ 1184934651e-1], [102., .8919017388e-2], [103., .6485415404e-2], [104., .\ 4539604143e-2], [105., .3045769172e-2], [106., .1948431718e-2], [107., .\ 1180610921e-2], [108., .6718343907e-3], [109., .3550212281e-3], [110., .\ 1715415292e-3], [111., .7412778757e-4], [112., .2769676342e-4], [113., .\ 8459415642e-5], [114., .1896654697e-5], [115., .2370818372e-6]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [87.82680806, 7.485382703, -.1585305596, 2.834595481] Here is a plot + HHH 0.05 HH HH + HH HH + HH H + H H 0.04 H H + HH H + H H 0.03 H H + HH H + H H + H H 0.02 HH HH + H HH + H H + HH H 0.01 HH HH + H H + HH HH + HHH HHH +****************************+-+-+-+-+-+-+-+-+-+-++-+-+-+-+-+-+-+-+*********- 0 40 50 60 70 80 90 100 110 --------------------------------------------- The occupants of cell, [1, 40], in a standard Young tableau of shape, [n, n, n], are all the integers from, 40, to , 118 as n goes to infinity, the distribution is [[40., .7081951736e-16], [41., .1796397513e-14], [42., .2333784127e-13], [43., .2069188920e-12], [44., .1407728551e-11], [45., .7834337208e-11], [46., .\ 3713177004e-10], [47., .1540847753e-9], [48., .5711948960e-9], [49., .\ 1920666751e-8], [50., .5928688982e-8], [51., .1696199709e-7], [52., .4533416534\ e-7], [53., .1139335791e-6], [54., .2707442202e-6], [55., .6112251610e-6], [56. , .1316282381e-5], [57., .2713587662e-5], [58., .5372006706e-5], [59., .\ 1024045292e-4], [60., .1884297935e-4], [61., .3354063918e-4], [62., .5786685157\ e-4], [63., .9693610694e-4], [64., .1579155489e-3], [65., .2505354878e-3], [66. , .3875990760e-3], [67., .5854347399e-3], [68., .8642197165e-3], [69., .\ 1248091819e-2], [70., .1764962185e-2], [71., .2445946818e-2], [72., .3324351043\ e-2], [73., .4434169406e-2], [74., .5808103195e-2], [75., .7475146453e-2], [76. , .9457845014e-2], [77., .1176938662e-1], [78., .1441072736e-1], [79., .\ 1736799456e-1], [80., .2061042298e-1], [81., .2408907568e-1], [82., .2773657135\ e-1], [83., .3146798528e-1], [84., .3518301614e-1], [85., .3876941880e-1], [86. , .4210760281e-1], [87., .4507619560e-1], [88., .4755827712e-1], [89., .\ 4944791770e-1], [90., .5065659980e-1], [91., .5111908092e-1], [92., .5079826212\ e-1], [93., .4968866230e-1], [94., .4781816189e-1], [95., .4524776840e-1], [96. , .4206926962e-1], [97., .3840077840e-1], [98., .3438033437e-1], [99., .\ 3015790776e-1], [100., .2588633561e-1], [101., .2171188803e-1], [102., .\ 1776528015e-1], [103., .1415398154e-1], [104., .1095659960e-1], [105., .\ 8219919081e-2], [106., .5958878371e-2], [107., .4159397242e-2], [108., .\ 2783599244e-2], [109., .1776664644e-2], [110., .1074365088e-2], [111., .\ 6103048362e-3], [112., .3220273654e-3], [113., .1554086027e-3], [114., .\ 6709126868e-4], [115., .2504971997e-4], [116., .7647334870e-5], [117., .\ 1714185208e-5], [118., .2142731509e-6]] The limiting average, standard deviation up to the, 4, -th scaled-moment are [90.38143545, 7.608583493, -.1616837885, 2.837433330] Here is a plot 0.05 HHHH + HH HH + HH H + H HH 0.04 HH H + H H + H H + H H 0.03 HH H + H H + H H + HH H 0.02 H H + H H + HH H + HH H 0.01 HH HH + HH HH + HH HH + HHHH HHH +*****************************-+---+---+---+--+---+---+---+--+---+-*********- 0 40 60 80 100 ----------------------- This took, 23787.683, seconds.