The Limiting Sorting Probabilities of The entries in the first row vs. thos\ e not related to it in lower rows in a random Standard Young tableau of shape, [n, n, n], as n goes to infinity for i from 2 to, 30 By Shalosh B. Ekhad --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 2], vs. those in the, 2, -th row from j=1 to j=, 1, as n goes to infinity, are as follows [-.3333333333] The cut off is at j=, 1 The limits of the sorting probabilities of the cell, [1, 2], vs. those in the, 3, -th row from j=1 to j=, 1, as n goes to infinity, are as follows [-.9259259259] The cut off is at j=, 1 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 3], vs. those in the, 2, -th row from j=1 to j=, 2, as n goes to infinity, are as follows [.2592592593, -.6296296296] The cut off is at j=, 2 The limits of the sorting probabilities of the cell, [1, 3], vs. those in the, 3, -th row from j=1 to j=, 2, as n goes to infinity, are as follows [-.7283950617, -.9862825789] The cut off is at j=, 1 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 4], vs. those in the, 2, -th row from j=1 to j=, 3, as n goes to infinity, are as follows [.6296296296, -.1275720165, -.7558299040] The cut off is at j=, 2 The limits of the sorting probabilities of the cell, [1, 4], vs. those in the, 3, -th row from j=1 to j=, 3, as n goes to infinity, are as follows [-.4430727023, -.9231824417, -.9957323579] The cut off is at j=, 1 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 5], vs. those in the, 2, -th row from j=1 to j=, 4, as n goes to infinity, are as follows [.8271604938, .3058984911, -.3527409440, -.8236041254] The cut off is at j=, 3 The limits of the sorting probabilities of the cell, [1, 5], vs. those in the, 3, -th row from j=1 to j=, 4, as n goes to infinity, are as follows [-.1316364375, -.7851953462, -.9701265051, -.9982613310] The cut off is at j=, 1 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 6], vs. those in the, 2, -th row from j=1 to j=, 5, as n goes to infinity, are as follows [.9231824417, .6076817558, .6304933191e-1, -.4962601681, -.8649708999] The cut off is at j=, 4 The limits of the sorting probabilities of the cell, [1, 6], vs. those in the, 3, -th row from j=1 to j=, 5, as n goes to infinity, are as follows [.1574455114, -.5766679650, -.8999023410, -.9860078541, -.9991628631] The cut off is at j=, 2 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 7], vs. those in the, 2, -th row from j=1 to j=, 6, as n goes to infinity, are as follows [.9670781893, .7921048621, .4034445969, -.1177045053, -.5940345840, -.892400306\ 2] The cut off is at j=, 4 The limits of the sorting probabilities of the cell, [1, 7], vs. those in the, 3, -th row from j=1 to j=, 6, as n goes to infinity, are as follows [.3988043828, -.3247077286, -.7710890453, -.9471407822, -.9925853586, -.9995482\ 118] The cut off is at j=, 2 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 8], vs. those in the, 2, -th row from j=1 to j=, 7, as n goes to infinity, are as follows [.9862825789, .8952395468, .6448448652, .2282730664, -.2545678915, -.6640541435 , -.9116835300] The cut off is at j=, 5 The limits of the sorting probabilities of the cell, [1, 8], vs. those in the, 3, -th row from j=1 to j=, 7, as n goes to infinity, are as follows [.5858012042, -.6271021200e-1, -.5869725827, -.8658890392, -.9694791981, -.9957\ 080122, -.9997350625] The cut off is at j=, 2 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 9], vs. those in the, 2, -th row from j=1 to j=, 8, as n goes to infinity, are as follows [.9944114210, .9492963471, .7997708118, .5009021942, .8154105396e-1, -.36042147\ 73, -.7161708800, -.9258448914] The cut off is at j=, 6 The limits of the sorting probabilities of the cell, [1, 9], vs. those in the, 3, -th row from j=1 to j=, 8, as n goes to infinity, are as follows [.7228402135, .1819322958, -.3650796202, -.7352043850, -.9163708461, -.98115411\ 13, -.9973447373, -.9998344958] The cut off is at j=, 3 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 10], vs. those in the, 2, -th row from j=1 to j=, 9, as n goes to infinity, are as follows [.9977645684, .9762570069, .8920783304, .6944086403, .3681295880, -.4088790720e\ -1, -.4439736729, -.7561705807, -.9366035682] The cut off is at j=, 6 The limits of the sorting probabilities of the cell, [1, 10], vs. those in the, 3, -th row from j=1 to j=, 9, as n goes to infinity, are as follows [.8190380685, .3926458753, -.1288640149, -.5598789522, -.8230907833, -.94519843\ 29, -.9877376451, -.9982697291, -.9998913356] The cut off is at j=, 3 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 11], vs. those in the, 2, -th row from j=1 to j=, 10, as n goes to infinity, are as follows [.9991193754, .9891841239, .9439841655, .8213490407, .5878583025, .2488174270, -.1434140781, -.5111402147, -.7876456743, -.9450019891] The cut off is at j=, 7 The limits of the sorting probabilities of the cell, [1, 11], vs. those in the, 3, -th row from j=1 to j=, 10, as n goes to infinity, are as follows [.8842901366, .5630980833, .9998033228e-1, -.3540156941, -.6872643657, -.877572\ 8791, -.9626131282, -.9916789444, -.9988244492, -.9999257277] The cut off is at j=, 4 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 12], vs. those in the, 2, -th row from j=1 to j=, 11, as n goes to infinity, are as follows [.9996575349, .9951866717, .9718434303, .8996097442, .7432511681, .4849360880, .1428334451, -.2298437392, -.5660125985, -.8129300178, -.9517052702] The cut off is at j=, 8 The limits of the sorting probabilities of the cell, [1, 12], vs. those in the, 3, -th row from j=1 to j=, 11, as n goes to infinity, are as follows [.9273302812, .6942463374, .3058794865, -.1357714330, -.5151106730, -.772507184\ 5, -.9126957965, -.9736298535, -.9941553435, -.9991734829, -.9999475227] The cut off is at j=, 4 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 13], vs. those in the, 2, -th row from j=1 to j=, 12, as n goes to infinity, are as follows [.9998682826, .9979004673, .9862326796, .9454831569, .8463142149, .6623253776, .3880199183, .4909234262e-1, -.3032626605, -.6114807937, -.8335977028, -.957155\ 6885] The cut off is at j=, 9 The limits of the sorting probabilities of the cell, [1, 13], vs. those in the, 3, -th row from j=1 to j=, 12, as n goes to infinity, are as follows [.9550667038, .7911146365, .4804317838, .7763857408e-1, -.3190662887, -.6308994\ 866, -.8309465298, -.9361204835, -.9808722426, -.9957747144, -.9994019897, -.99\ 99618755] The cut off is at j=, 5 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 14], vs. those in the, 2, -th row from j=1 to j=, 13, as n goes to infinity, are as follows [.9999498220, .9991000011, .9934283548, .9712650665, .9111669405, .7871934314, .5816160815, .2980731184, -.3378064107e-1, -.3661166983, -.6496290305, -.850744\ 3565, -.9616575840] The cut off is at j=, 9 The limits of the sorting probabilities of the cell, [1, 14], vs. those in the, 3, -th row from j=1 to j=, 13, as n goes to infinity, are as follows [.9725931025, .8602769462, .6213526701, .2730946727, -.1140455409, -.4599513284 , -.7150470137, -.8719461691, -.9522101424, -.9857906832, -.9968697957, -.99955\ 66663, -.9999716386] The cut off is at j=, 5 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 15], vs. those in the, 2, -th row from j=1 to j=, 14, as n goes to infinity, are as follows [.9999810439, .9996200407, .9969287066, .9852471171, .9502135819, .8704970534, .7248084742, .5030341441, .2152754312, -.1071615203, -.4203332493, -.6819900362 , -.8651533191, -.9654265111] The cut off is at j=, 10 The limits of the sorting probabilities of the cell, [1, 15], vs. those in the, 3, -th row from j=1 to j=, 14, as n goes to infinity, are as follows [.9834831267, .9082657332, .7305734981, .4426787338, .8598396551e-1, -.27098008\ 74, -.5681699044, -.7770206123, -.9013292195, -.9635488780, -.9892251747, -.997\ 6318221, -.9996643900, -.9999784663] The cut off is at j=, 6 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 16], vs. those in the, 2, -th row from j=1 to j=, 15, as n goes to infinity, are as follows [.9999928914, .9998417378, .9985912361, .9926003151, .9728527258, .9236106762, .8251411940, .6611124022, .4277006286, .1393930363, -.1723124056, -.4674337149, -.7097112882, -.8773977647, -.9686189523] The cut off is at j=, 11 The limits of the sorting probabilities of the cell, [1, 16], vs. those in the, 3, -th row from j=1 to j=, 15, as n goes to infinity, are as follows [.9901517968, .9407600947, .8123502714, .5832800840, .2702096703, -.7661235041e\ -1, -.3980543151, -.6516743448, -.8232991170, -.9227985485, -.9717213560, -.991\ 6819500, -.9981753386, -.9997412793, -.9999833572] The cut off is at j=, 6 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 17], vs. those in the, 2, -th row from j=1 to j=, 16, as n goes to infinity, are as follows [.9999973517, .9999348666, .9993645226, .9963650882, .9855566258, .9561831272, .8922915072, .7766050027, .5975203573, .3562097216, .6999036653e-1, -.230345151\ 8, -.5086253086, -.7336659120, -.8879056258, -.9713509572] The cut off is at j=, 12 The limits of the sorting probabilities of the cell, [1, 17], vs. those in the, 3, -th row from j=1 to j=, 16, as n goes to infinity, are as follows [.9941837808, .9623039574, .8717922900, .6954306738, .4318242899, .1116421050, -.2150597209, -.5005981454, -.7165626304, -.8583279472, -.9387627789, -.9777298\ 517, -.9934767654, -.9985714344, -.9997973488, -.9999869347] The cut off is at j=, 7 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 18], vs. those in the, 2, -th row from j=1 to j=, 17, as n goes to infinity, are as follows [.9999990192, .9999734810, .9997176120, .9982476199, .9924838221, .9754926520, .9354902292, .8571864751, .7261498026, .5350172002, .2888141470, .6549186603e-2 , -.2822197690, -.5448725343, -.7545281303, -.8970021062, -.9737102098] The cut off is at j=, 13 The limits of the sorting probabilities of the cell, [1, 18], vs. those in the, 3, -th row from j=1 to j=, 17, as n goes to infinity, are as follows [.9965943589, .9763276082, .9139055601, .7819588418, .5678287886, .2849244041, -.3001553964e-1, -.3316951461, -.5834110303, -.7673966685, -.8851850644, -.9508\ 231452, -.9822259452, -.9948125676, -.9988656110, -.9998390141, -.9999896004] The cut off is at j=, 7 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 19], vs. those in the, 2, -th row from j=1 to j=, 18, as n goes to infinity, are as follows [.9999996386, .9999893071, .9998762040, .9991693977, .9961663080, .9866026540, .9623350385, .9112124174, .8191992712, .6747843562, .4742609808, .2255510150, - .5146597813e-1, -.3287569944, -.5769518121, -.7728255741, -.9049382490, -.97576\ 40600] The cut off is at j=, 13 The limits of the sorting probabilities of the cell, [1, 19], vs. those in the, 3, -th row from j=1 to j=, 18, as n goes to infinity, are as follows [.9980212487, .9853098575, .9430804943, .8468139860, .6782412178, .4376081359, .1475232711, -.1544021555, -.4294680991, -.6504993259, -.8075611121, -.90602689\ 22, -.9600660681, -.9856437393, -.9958232872, -.9990877903, -.9998704975, -.999\ 9916195] The cut off is at j=, 8 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 20], vs. those in the, 2, -th row from j=1 to j=, 19, as n goes to infinity, are as follows [.9999998675, .9999957261, .9999463928, .9996123348, .9980799609, .9928271649, .9785138123, .9461884382, .8838755821, .7791451806, .6232893066, .4156690583, .\ 1663254974, -.1045732480, -.3706554832, -.6054929892, -.7889761959, -.911910578\ 8, -.9775650003] The cut off is at j=, 14 The limits of the sorting probabilities of the cell, [1, 20], vs. those in the, 3, -th row from j=1 to j=, 19, as n goes to infinity, are as follows [.9988583619, .9909813442, .9628968724, .8942062829, .7651096870, .5671404388, .3103307940, .2196422402e-1, -.2625085742, -.5113054204, -.7050911234, -.839566\ 6299, -.9223836605, -.9672427298, -.9882788682, -.9965994029, -.9992581261, -.9\ 998946451, -.9999931714] The cut off is at j=, 9 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 21], vs. those in the, 2, -th row from j=1 to j=, 20, as n goes to infinity, are as follows [.9999999516, .9999983054, .9999770457, .9998215982, .9990542994, .9962322728, .9880011675, .9681389112, .9272776905, .8540257424, .7377267564, .5722526811, .\ 3594852199, .1109657310, -.1532564047, -.4085093369, -.6310109066, -.8033147485 , -.9180748853, -.9791544815] The cut off is at j=, 15 The limits of the sorting probabilities of the cell, [1, 21], vs. those in the, 3, -th row from j=1 to j=, 20, as n goes to infinity, are as follows [.9993455437, .9945169930, .9761228565, .9280695168, .8315941662, .6734467173, .4538714408, .1894636029, -.9104835934e-1, -.3559196754, -.5798480658, -.749743\ 5583, -.8652829967, -.9353553582, -.9728816428, -.9903366851, -.9972033312, -.9\ 993904848, -.9999134163, -.9999943802] The cut off is at j=, 9 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 22], vs. those in the, 2, -th row from j=1 to j=, 21, as n goes to infinity, are as follows [.9999999824, .9999993330, .9999902716, .9999189549, .9995412875, .9980552869, .9934291843, .9815364079, .9555060711, .9058982123, .8221875823, .6955306507, .\ 5221061049, .3058301903, .5925844838e-1, -.1979578547, -.4428241945, -.65392950\ 29, -.8161120687, -.9235560736, -.9805656221] The cut off is at j=, 16 The limits of the sorting probabilities of the cell, [1, 22], vs. those in the, 3, -th row from j=1 to j=, 21, as n goes to infinity, are as follows [.9996270214, .9966960651, .9848131525, .9517875935, .8812479440, .7581744338, .5761302274, .3422415636, .7691991936e-1, -.1917711128, -.4363964435, -.6373696\ 464, -.7864671561, -.8861113186, -.9457427388, -.9773606632, -.9919624041, -.99\ 76789460, -.9994945915, -.9999281859, -.9999953328] The cut off is at j=, 10 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 23], vs. those in the, 2, -th row from j=1 to j=, 22, as n goes to infinity, are as follows [.9999999936, .9999997392, .9999959158, .9999636182, .9997806226, .9990123353, .9964660972, .9895104720, .9733565244, .9407227468, .8823781676, .7888416459, .\ 6530353066, .4731573404, .2547388726, .1097181715e-1, -.2390771006, -.474031291\ 6, -.6746003268, -.8275893178, -.9284553219, -.9818251665] The cut off is at j=, 17 The limits of the sorting probabilities of the cell, [1, 23], vs. those in the, 3, -th row from j=1 to j=, 22, as n goes to infinity, are as follows [.9997885802, .9980252677, .9904435010, .9681056429, .9175308260, .8239688138, .6771287056, .4767237387, .2348225545, -.2645739053e-1, -.2809776764, -.5056597\ 565, -.6857785066, -.8168392727, -.9031092010, -.9541358213, -.9809539528, -.99\ 32603927, -.9980576156, -.9995773850, -.9999399356, -.9999960919] The cut off is at j=, 10 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 24], vs. those in the, 2, -th row from j=1 to j=, 23, as n goes to infinity, are as follows [.9999999977, .9999998987, .9999983002, .9999838466, .9998964456, .9995058443, .9981309356, .9941492399, .9843606062, .9634511238, .9239501823, .8570518683, .\ 7544139460, .6106234027, .4256177467, .2061874513, -.3413016377e-1, -.276972147\ 3, -.5024994283, -.6933168515, -.8379286353, -.9328553590, -.9829549215] The cut off is at j=, 17 The limits of the sorting probabilities of the cell, [1, 24], vs. those in the, 3, -th row from j=1 to j=, 23, as n goes to infinity, are as follows [.9998807525, .9988285107, .9940453228, .9791532582, .9435287765, .8738847004, .7583224393, .5914256449, .3779644166, .1332096278, -.1205225758, -.3596820115, -.5652890593, -.7266541281, -.8420986211, -.9170812258, -.9609743763, -.9838631\ 598, -.9943067790, -.9983621153, -.9996438947, -.9999493772, -.9999967027] The cut off is at j=, 11 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 25], vs. those in the, 2, -th row from j=1 to j=, 24, as n goes to infinity, are as follows [.9999999992, .9999999609, .9999992983, .9999929007, .9999517075, .9997561723, .9990267066, .9967918970, .9909885298, .9779095170, .9518628808, .9053820212, .\ 8302423636, .7192733254, .5685956382, .3796246411, .1601129412, -.7627881886e-1 , -.3119623615, -.5285450160, -.7103256219, -.8472812042, -.9368244112, -.98397\ 28256] The cut off is at j=, 18 The limits of the sorting probabilities of the cell, [1, 25], vs. those in the, 3, -th row from j=1 to j=, 24, as n goes to infinity, are as follows [.9999330489, .9993097956, .9963230739, .9865245792, .9618306098, .9109688460, .8220200935, .6865390700, .5036008926, .2818294534, .3827416870e-1, -.205558816\ 7, -.4289732860, -.6166841202, -.7612940731, -.8632207407, -.9286443843, -.9665\ 897667, -.9862384724, -.9951578497, -.9986092153, -.9996978178, -.9999570340, -\ .9999971987] The cut off is at j=, 12 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 26], vs. those in the, 2, -th row from j=1 to j=, 25, as n goes to infinity, are as follows [.9999999997, .9999999850, .9999997125, .9999969094, .9999777315, .9998812367, .9995004559, .9982687193, .9948965724, .9868941992, .9701289130, .9386742544, .\ 8852276779, .8022509476, .6837333508, .5271840754, .3352591118, .1164271816, -.\ 1156955852, -.3443319362, -.5524404486, -.7258349991, -.8557734247, -.940419188\ 1, -.9848937551] The cut off is at j=, 19 The limits of the sorting probabilities of the cell, [1, 26], vs. those in the, 3, -th row from j=1 to j=, 25, as n goes to infinity, are as follows [.9999625703, .9995959326, .9977484550, .9913782909, .9745095785, .9380025959, .8709031917, .7634375781, .6107310150, .4156907627, .1896608346, -.4962415623e-\ 1, -.2820943526, -.4899221236, -.6610599156, -.7907605996, -.8809769607, -.9382\ 755754, -.9712342380, -.9881930589, -.9958557422, -.9988114241, -.9997419080, -\ .9999632962, -.9999976049] The cut off is at j=, 12 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 27], vs. those in the, 2, -th row from j=1 to j=, 26, as n goes to infinity, are as follows [.9999999999, .9999999942, .9999998830, .9999986663, .9999898391, .9999428477, .9997470539, .9990795038, .9971561452, .9923586114, .9818043801, .9610262614, .\ 9239954037, .8637001273, .7733515889, .6480566255, .4865642091, .2925600055, .\ 7502678420e-1, -.1525891664, -.3743334887, -.5744210534, -.7400220684, -.863511\ 6886, -.9436871705, -.9857301390] The cut off is at j=, 20 The limits of the sorting probabilities of the cell, [1, 27], vs. those in the, 3, -th row from j=1 to j=, 26, as n goes to infinity, are as follows [.9999791573, .9997648367, .9986319094, .9945358795, .9831657972, .9573732350, .9076757460, .8242032879, .6997342057, .5327568107, .3292506946, .1022999983, -\ .1304725011, -.3507808067, -.5435311421, -.6994577315, -.8159225074, -.89597997\ 30, -.9463462791, -.9751016644, -.9898131157, -.9964323649, -.9989781824, -.999\ 7782409, -.9999684577, -.9999979400] The cut off is at j=, 13 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 28], vs. those in the, 2, -th row from j=1 to j=, 27, as n goes to infinity, are as follows [1.000000000, .9999999978, .9999999527, .9999994292, .9999954091, .9999728060, .9998735324, .9995173644, .9984391088, .9956166776, .9891074498, .9756840314, .\ 9506379240, .9079539863, .8410074014, .7437887033, .6124601851, .4468654618, .2\ 515348036, .3580014681e-1, -.1871542101, -.4021915377, -.5946908579, -.75303813\ 55, -.8705861066, -.9466683791, -.9864924336] The cut off is at j=, 21 The limits of the sorting probabilities of the cell, [1, 28], vs. those in the, 3, -th row from j=1 to j=, 27, as n goes to infinity, are as follows [.9999884365, .9998638848, .9991746672, .9965674361, .9889971561, .9710365089, .9348397869, .8712335006, .7719561232, .6324492402, .4541680949, .2454134902, .\ 2021022922e-1, -.2044825408, -.4123152826, -.5907122395, -.7327631661, -.837490\ 8632, -.9087188923, -.9531480355, -.9783424100, -.9911649567, -.9969121377, -.9\ 991166937, -.9998083984, -.9999727429, -.9999982185] The cut off is at j=, 14 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 29], vs. those in the, 2, -th row from j=1 to j=, 28, as n goes to infinity, are as follows [1.000000000, .9999999992, .9999999810, .9999997576, .9999979448, .9999871972, .9999375166, .9997502368, .9991553898, .9975238083, .9935850413, .9850898452, .\ 9685222899, .9390223901, .8906869910, .8173470090, .7137770969, .5771211761, .4\ 081800836, .2121680188, -.1367653016e-2, -.2195709527, -.4281057336, -.61342737\ 67, -.7650131284, -.8770734447, -.9493967572, -.9871894915] The cut off is at j=, 21 The limits of the sorting probabilities of the cell, [1, 29], vs. those in the, 3, -th row from j=1 to j=, 28, as n goes to infinity, are as follows [.9999936067, .9999216158, .9995054157, .9978612761, .9928776304, .9805362280, .9545752292, .9069497288, .8293184255, .7153248803, .5629747042, .3762068142, .\ 1649608781, -.5642113462e-1, -.2720027326, -.4673923262, -.6322795334, -.761726\ 0704, -.8560486733, -.9195862018, -.9589113028, -.9810740889, -.9923000767, -.9\ 973139338, -.9992325101, -.9998335986, -.9999763243, -.9999984515] The cut off is at j=, 14 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 30], vs. those in the, 2, -th row from j=1 to j=, 29, as n goes to infinity, are as follows [1.000000000, .9999999997, .9999999924, .9999998978, .9999990879, .9999940324, .9999694726, .9998723313, .9995490580, .9986211947, .9962797439, .9910054193, .\ 9802699131, .9603290148, .9262541310, .8723344740, .7929025609, .6835032422, .5\ 421823546, .3705705756, .1744276414, -.3659149152e-1, -.2500054642, -.452253787\ 6, -.6307855916, -.7760591471, -.8830394556, -.9519012603, -.9878288521] The cut off is at j=, 22 The limits of the sorting probabilities of the cell, [1, 30], vs. those in the, 3, -th row from j=1 to j=, 29, as n goes to infinity, are as follows [.9999964766, .9999550756, .9997054600, .9986775811, .9954308821, .9870542433, .9686965259, .9336070751, .8739952288, .7827244046, .6554514212, .4925128767, .\ 2998297589, .8839426636e-1, -.1276015658, -.3334570506, -.5166757928, -.6689510\ 749, -.7869799157, -.8720751378, -.9288984900, -.9638195166, -.9833894367, -.99\ 32588123, -.9976524734, -.9993299513, -.9998547881, -.9999793363, -.9999986476] The cut off is at j=, 15 ------------------------- This ends this article that took, 5780.369, seconds to produce ----------------------- This took, 5780.370, seconds.