The trends in the Expectation, Variance and moemnts up to the, 4, of the Dura\ tion of the Dym-Luks Ball and Cell Game with r balls and n cells for fi\ xed n from 1 to, 6, as r grows larger By Shalosh B. Ekhad Suppose that you have r balls and (uniformly at) randomly throw them into, 2, cells. Whenever a ball has no cell-mates, it is removed from the game an\ d this continues until all the balls are removed The sequence of the expectation for balls from r=1 to r= , 30, are [1., 2., 3.333333333, 5.333333333, 8.533333333, 13.86666667, 23.00952381, 39.00952381, 67.45396825, 118.6539683, 211.7448773, 382.4115440, 697.4884671, 1282.631324, 2374.897991, 4422.897991, 8277.956814, 15559.73459, 29356.78722, 55571.18722, 105503.3777, 200828.4686, 7 7 383189.5121, 732714.8454, 0.1403803485 10 , 0.2694358562 10 , 7 7 8 8 0.5179872044 10 , 0.9973362330 10 , 0.1922975736 10 , 0.3712545443 10 ] The sequence of E_n(r), is [0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0. , 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.] The sequence of the standard-deviations for number of balls up to , 30, is [0., 1.414213562, 1.563471920, 2.108185107, 3.388870672, 5.881798665, 10.44243914, 18.68273361, 33.61080943, 60.82702122, 110.7824578, 203.0497412, 374.4163297, 694.2583155, 1293.811738, 2422.024941, 4552.343166, 8587.248345, 16250.71379, 30842.40996, 58689.24914, 7 111942.8676, 213977.9741, 409822.1351, 786329.1130, 0.1511239421 10 , 7 7 8 8 0.2908886277 10 , 0.5607063733 10 , 0.1082219957 10 , 0.2091353530 10 ] The sequence of scaled, 3, -th moments are [0., 2.121320343, 1.763751381, 1.359779394, 1.304153586, 1.347349293, 1.370932102, 1.380297735, 1.386113428, 1.391890466, 1.398040736, 1.404230798, 1.410130758, 1.415555970, 1.420442114, 1.424796689, 1.428662150, 1.432093679, 1.435147476, 1.437875361, 1.440322831, 1.442528821, 1.444526231, 1.446342714, 1.448001501, 1.449522143, 1.450921164, 1.452212586, 1.453408374, 1.454518789] --------------------------------------------- The sequence of scaled, 4, -th moments are [0., 9.500000000, 7.623966942, 5.715000000, 5.714629921, 5.996101181, 6.104628860, 6.141670238, 6.165628272, 6.190727217, 6.217677043, 6.244722574, 6.270424257, 6.294028624, 6.315291955, 6.334263183, 6.351130633, 6.366131910, 6.379506891, 6.391476223, 6.402233645, 6.411945184, 6.420751399, 6.428770705, 6.436102802, 6.442831809, 6.449028959, 6.454754850, 6.460061287, 6.464992785] --------------------------------------------- Suppose that you have r balls and (uniformly at) randomly throw them into, 3, cells. Whenever a ball has no cell-mates, it is removed from the game an\ d this continues until all the balls are removed The sequence of the expectation for balls from r=1 to r= , 30, are [1., 1.500000000, 2.250000000, 3.150000000, 4.200000000, 5.505555556, 7.166064587, 9.329299601, 12.19903396, 16.06080809, 21.31669879, 28.53537096, 38.52397352, 52.43158599, 71.89828688, 99.27027378, 137.9107211, 192.6495453, 270.4349110, 381.2780000, 539.6244646, 766.3471991, 1091.644570, 1559.259188, 2232.623970, 3203.822973, 4606.665823, 6635.777687, 9574.491333, 13835.62605] The sequence of E_n(r), is [0., -.2500000000, -.2500000000, -.1937500000, -.1562500000, -.1163194444, -.\ 8304255610e-1, -.5554972888e-1, -.3347162452e-1, -.1603342521e-1, -.2419005672e\ -2, .8123277604e-2, .1623830806e-1, .2245744000e-1, .2720763762e-1, .3082639037\ e-1, .3357747934e-1, .3566543411e-1, .3724791452e-1, .3844591399e-1, .\ 3935196897e-1, .4003666024e-1, .4055370479e-1, .4094391216e-1, .4123824013e-1, .4146014392e-1, .4162737578e-1, .4175335960e-1, .4184823846e-1, .4191967130e-1] The sequence of the standard-deviations for number of balls up to , 30, is [0., 0.8660254040, 1.060660172, 1.227802916, 1.425657743, 1.688678715, 2.086498189, 2.694810544, 3.607700676, 4.945368719, 6.871801948, 9.618723514, 13.51761633, 19.04418425, 26.88176700, 38.01277313, 53.85100822, 76.43349924, 108.6988605, 154.8916217, 221.1500405, 316.3613993, 453.4075328, 650.9800622, 936.2279143, 1348.621513, 1945.596634, 2810.802955, 4066.166951, 5889.543531] The sequence of scaled, 3, -th moments are [0., 2.309401077, 1.355287997, 1.390057309, 1.115753528, 0.9864738297, 0.9279097732, 0.9409349655, 0.9839390123, 1.025155207, 1.053691849, 1.070672462, 1.080199503, 1.085708284, 1.089345248, 1.092261799, 1.094999008, 1.097765719, 1.100607089, 1.103497813, 1.106390070, 1.109236049, 1.111996958, 1.114645284, 1.117163978, 1.119544508, 1.121784727, 1.123886958, 1.125856455, 1.127700231] --------------------------------------------- The sequence of scaled, 4, -th moments are [0., 10.33333333, 6.388888889, 5.735080815, 4.951380473, 4.484912733, 4.358999224, 4.463130346, 4.640396366, 4.787196108, 4.880022216, 4.931705543, 4.959554593, 4.975692516, 4.986938198, 4.996645035, 5.006239531, 5.016175926, 5.026464630, 5.036945810, 5.047422033, 5.057716314, 5.067692294, 5.077256371, 5.086352200, 5.094952810, 5.103052836, 5.110661875, 5.117799222, 5.124489939] --------------------------------------------- Suppose that you have r balls and (uniformly at) randomly throw them into, 4, cells. Whenever a ball has no cell-mates, it is removed from the game an\ d this continues until all the balls are removed The sequence of the expectation for balls from r=1 to r= , 30, are [1., 1.333333333, 1.866666667, 2.488888889, 3.152592593, 3.887738928, 4.731939313, 5.708647102, 6.858749575, 8.229075218, 9.879790143, 11.88680728, 14.34659749, 17.38202271, 21.14990725, 25.85080667, 31.74163566, 39.15200649, 48.50538540, 60.34650872, 75.37693609, 94.50118748, 118.8866535, 150.0414390, 189.9155696, 241.0326454, 306.6611984, 391.0378433, 499.6580220, 639.6550024] The sequence of E_n(r), is [0., -.3333333333, -.3925925926, -.3629629630, -.3313580247, -.2985436511, -.25\ 70083645, -.2167431911, -.1764986328, -.1380024878, -.1016263468, -.6768994504e\ -1, -.3630679553e-1, -.7480796787e-2, .1885805275e-1, .4282534758e-1, .\ 6456305531e-1, .8422634194e-1, .1019746733, .1179659209, .1323525714, .14527940\ 23, .1568821705, .1672869966, .1766102174, .1849585510, .1924294601, .199111638\ 7, .2050855663, .2104240930] The sequence of the standard-deviations for number of balls up to , 30, is [0., 0.6666666667, 0.8743251363, 0.9723756492, 1.074685905, 1.203910294, 1.337125830, 1.506467606, 1.721854261, 2.006503461, 2.385390827, 2.888838306, 3.552957312, 4.421809246, 5.550352117, 7.008282557, 8.884850847, 11.29491159, 14.38661903, 18.35131427, 23.43630809, 29.96146243, 38.34073785, 49.11022255, 62.96461765, 80.80475647, 103.7995257, 133.4665899, 171.7776778, 221.2959618] The sequence of scaled, 3, -th moments are [0., 2.500000000, 1.253680107, 1.182845567, 1.117885348, 0.9480372635, 0.8897867934, 0.8156911199, 0.7746360159, 0.7577513231, 0.7652910500, 0.7886267157, 0.8181443901, 0.8465233780, 0.8700578474, 0.8878761466, 0.9006266676, 0.9094768216, 0.9155781750, 0.9198620033, 0.9230067209, 0.9254740197, 0.9275617848, 0.9294526537, 0.9312520664, 0.9330157954, 0.9347689355, 0.9365185473, 0.9382618395, 0.9399912985] --------------------------------------------- The sequence of scaled, 4, -th moments are [0., 11.25000000, 5.752163332, 5.574179379, 4.839998599, 4.488368589, 4.217174328, 4.040797531, 3.940042588, 3.923963375, 3.972807963, 4.058382541, 4.151917347, 4.234759279, 4.299542025, 4.346417274, 4.378827126, 4.400819165, 4.415860088, 4.426530218, 4.434599246, 4.441212800, 4.447076138, 4.452600012, 4.458005905, 4.463398073, 4.468811216, 4.474241203, 4.479664395, 4.485049422] --------------------------------------------- Suppose that you have r balls and (uniformly at) randomly throw them into, 5, cells. Whenever a ball has no cell-mates, it is removed from the game an\ d this continues until all the balls are removed The sequence of the expectation for balls from r=1 to r= , 30, are [1., 1.250000000, 1.666666667, 2.157738095, 2.665321265, 3.191543257, 3.763378585, 4.393444064, 5.090514993, 5.872639013, 6.756883666, 7.764713196, 8.921408408, 10.25729223, 11.80876766, 13.61964984, 15.74276478, 18.24189434, 21.19414357, 24.69282609, 28.85098409, 33.80568513, 39.72327054, 46.80576911, 55.29873861, 65.50085722, 77.77566026, 92.56590790, 110.4111804, 131.9694343] The sequence of E_n(r), is [0., -.3750000000, -.4791666667, -.4763764881, -.4570745678, -.4394788789, -.41\ 26003026, -.3785812717, -.3437841737, -.3067182134, -.2691304464, -.2314285973, -.1941114801, -.1575058000, -.1218548734, -.8732953358e-1, -.5404617147e-1, -.\ 2207888692e-1, .8530799478e-2, .3776637420e-1, .6563037763e-1, .9213972387e-1, .1173220236, .1412126995, .1638527275, .1852868734, .2055623193, .2247275979, .\ 2428317703, .2599237942] The sequence of the standard-deviations for number of balls up to , 30, is [0., 0.5590169942, 0.7682953714, 0.8549751441, 0.9155075842, 0.9995215844, 1.085514770, 1.170198962, 1.271239698, 1.387581671, 1.527874015, 1.699691707, 1.912564355, 2.177517347, 2.507309593, 2.916681919, 3.422784927, 4.045786564, 4.809665068, 5.743188580, 6.881096314, 8.265514290, 9.947656957, 11.98988374, 14.46819685, 17.47528632, 21.12425132, 25.55315494, 30.93060472, 37.46259463] The sequence of scaled, 3, -th moments are [0., 2.683281572, 1.266491055, 1.000306120, 1.047031873, 0.9476581321, 0.8314972696, 0.7999043838, 0.7432948413, 0.7052136728, 0.6759361629, 0.6582847997, 0.6527910440, 0.6584472190, 0.6727172189, 0.6922063926, 0.7135922938, 0.7343022199, 0.7527559407, 0.7682621238, 0.7807572401, 0.7905368393, 0.7980503063, 0.8037716224, 0.8081313971, 0.8114894320, 0.8141304168, 0.8162709239, 0.8180706536, 0.8196442163] --------------------------------------------- The sequence of scaled, 4, -th moments are [0., 12.20000000, 5.462283737, 5.179050755, 4.875654509, 4.399519337, 4.205401999, 4.017857151, 3.883734672, 3.788630259, 3.723447142, 3.693092890, 3.694270607, 3.722439717, 3.769329353, 3.825563524, 3.882870166, 3.935509627, 3.980493846, 4.017012927, 4.045616043, 4.067502482, 4.084045048, 4.096528198, 4.106035942, 4.113426909, 4.119352194, 4.124289494, 4.128579701, 4.132459776] --------------------------------------------- Suppose that you have r balls and (uniformly at) randomly throw them into, 6, cells. Whenever a ball has no cell-mates, it is removed from the game an\ d this continues until all the balls are removed The sequence of the expectation for balls from r=1 to r= , 30, are [1., 1.200000000, 1.542857143, 1.954285714, 2.377280551, 2.799230040, 3.238184533, 3.708682197, 4.213473516, 4.758162316, 5.352496057, 6.004536219, 6.724200556, 7.522876263, 8.413538523, 9.411234415, 10.53338362, 11.80021727, 13.23526556, 14.86593164, 16.72416296, 18.84723884, 21.27869535, 24.06941231, 27.27889182, 30.97676262, 35.24455115, 40.17776714, 45.88836077, 52.50761808] The sequence of E_n(r), is [0., -.4000000000, -.5371428571, -.5577142857, -.5494394492, -.5422099603, -.52\ 98246096, -.5072245463, -.4801906673, -.4514799024, -.4200312904, -.3871647703, -.3533543139, -.3189157887, -.2841991713, -.2494421280, -.2148473908, -.1805754\ 622, -.1467552390, -.1134891592, -.8085783690e-1, -.4892376642e-1, -.1773438493\ e-1, .1267538357e-1, .4228100412e-1, .7106655222e-1, .9902322262e-1, .126148078\ 6, .1524430027, .1779138174] The sequence of the standard-deviations for number of balls up to , 30, is [0., 0.4898979486, 0.6963461490, 0.7852102278, 0.8275051903, 0.8828373720, 0.9498553079, 1.009855988, 1.070155228, 1.139426699, 1.215168642, 1.301315325, 1.401357093, 1.518723943, 1.657880849, 1.823765617, 2.022037462, 2.259087142, 2.542131968, 2.879341408, 3.280011706, 3.754791074, 4.315956674, 4.977744694, 5.756736982, 6.672310902, 7.747162305, 9.007914855, 10.48583220, 12.21765291] The sequence of scaled, 3, -th moments are [0., 2.857738034, 1.324166040, 0.8977698088, 0.9256003405, 0.9395293351, 0.8196825919, 0.7523213289, 0.7310752928, 0.6865983415, 0.6532354232, 0.6275188330, 0.6053925932, 0.5902539253, 0.5816717884, 0.5799208032, 0.5844738226, 0.5942633714, 0.6078241488, 0.6235632524, 0.6400168243, 0.6560228580, 0.6707882330, 0.6838694793, 0.6951041091, 0.7045265473, 0.7122908366, 0.7186102004, 0.7237151951, 0.7278278534] --------------------------------------------- The sequence of scaled, 4, -th moments are [0., 13.16666667, 5.368992960, 4.776249698, 4.829470759, 4.408193786, 4.119176525, 4.027712290, 3.879799414, 3.771737055, 3.699126398, 3.632360175, 3.587981307, 3.559454383, 3.547985847, 3.552463325, 3.570770345, 3.599752597, 3.635624399, 3.674614429, 3.713508402, 3.749961040, 3.782551352, 3.810655063, 3.834233400, 3.853616952, 3.869327730, 3.881952802, 3.892065393, 3.900182369] --------------------------------------------- ----------------------------- This ends this paper that took, 82.694, to produce.