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], as n goes to infinity for i from 2 to, 60 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 [-.5000000000] 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 [0., -.7500000000] 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 [.3750000000, -.3750000000, -.8437500000] The cut off is at j=, 2 --------------------------------------------- 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 [.6250000000, 0., -.5625000000, -.8906250000] The cut off is at j=, 2 --------------------------------------------- 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 [.7812500000, .3125000000, -.2343750000, -.6718750000, -.9179687500] The cut off is at j=, 3 --------------------------------------------- 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 [.8750000000, .5468750000, .7812500000e-1, -.3906250000, -.7421875000, -.935546\ 8750] The cut off is at j=, 4 --------------------------------------------- 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 [.9296875000, .7109375000, .3417968750, -.9765625000e-1, -.5004882812, -.790527\ 3438, -.9476318359] The cut off is at j=, 4 --------------------------------------------- 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 [.9609375000, .8203125000, .5468750000, .1708984375, -.2319335938, -.5810546875 , -.8254394531, -.9563598633] The cut off is at j=, 5 --------------------------------------------- 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 [.9785156250, .8906250000, .6972656250, .3964843750, .2990722656e-1, -.33666992\ 19, -.6421508789, -.8516235352, -.9629058838] The cut off is at j=, 6 --------------------------------------------- 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 [.9882812500, .9345703125, .8027343750, .5742187500, .2631835938, -.8673095703e\ -1, -.4199829102, -.6897583008, -.8718566895, -.9679641724] The cut off is at j=, 6 --------------------------------------------- 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 [.9936523438, .9614257812, .8741455078, .7075195312, .4575805664, .1465454102, -.1839294434, -.4874267578, -.7276954651, -.8878746033, -.9719686508] The cut off is at j=, 7 --------------------------------------------- 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 [.9965820312, .9775390625, .9211425781, .8036499023, .6113891602, .3499145508, .4486083984e-1, -.2656402588, -.5428733826, -.7584991455, -.9008121490, -.97520\ 30373] The cut off is at j=, 8 --------------------------------------------- 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 [.9981689453, .9870605469, .9513549805, .8707885742, .7281188965, .5180053711, .2518615723, -.4385375977e-1, -.3349485397, -.5890789032, -.7839121819, -.91143\ 94188, -.9778598547] The cut off is at j=, 8 --------------------------------------------- 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 [.9990234375, .9926147461, .9703979492, .9164428711, .8137207031, .6510772705, .4292907715, .1631469727, -.1214790344, -.3942456245, -.6280455589, -.805166721\ 3, -.9202954769, -.9800738692] The cut off is at j=, 9 --------------------------------------------- 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 [.9994812012, .9958190918, .9822006226, .9467926025, .8747119904, .7527294159, .5748381615, .3461208344, .8309590816e-1, -.1896706820, -.4453893602, -.6612557\ 769, -.8231555894, -.9277677760, -.9819419440] The cut off is at j=, 10 --------------------------------------------- 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 [.9997253418, .9976501465, .9894104004, .9665794373, .9171123505, .8280715942, .6899442673, .5008413792, .2687605619, .1089298725e-1, -.2498397827, -.48983244\ 60, -.6898263320, -.8385397345, -.9341412075, -.9835353019] The cut off is at j=, 11 --------------------------------------------- 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 [.9998550415, .9986877441, .9937591553, .9792633057, .9459681511, .8824853897, .7781922817, .6269099712, .4299277961, .1971306801, -.5429020524e-1, -.30317148\ 57, -.5287201460, -.7146118991, -.8518177168, -.9396294402, -.9849073600] The cut off is at j=, 11 --------------------------------------------- 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 [.9999237061, .9992713928, .9963531494, .9872741699, .9652442932, .9209091663, .8440616131, .7264378071, .5647050738, .3625391573, .1309672892, -.1132357717, -.3506554142, -.5629633637, -.7362759756, -.8633718910, -.9443955370, -.9860988\ 843] The cut off is at j=, 12 --------------------------------------------- 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 [.9999599457, .9995975494, .9978852272, .9922676086, .9779114723, .9475102425, .8920913339, .8028932810, .6738746688, .5040552989, .2988568936, .6991652399e-1 , -.1666551912, -.3931170041, -.5932930708, -.7553403629, -.8734998467, -.94856\ 58717, -.9871414679] The cut off is at j=, 13 --------------------------------------------- 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 [.9999790192, .9997787476, .9987821579, .9953439236, .9861149788, .9656062126, .9263982773, .8602348864, .7598870769, .6213115305, .4454271831, .2388963206, .\ 1358992513e-1, -.2151827225, -.4312457787, -.6203009528, -.7722202892, -.882436\ 2783, -.9522397380, -.9880599345] The cut off is at j=, 14 --------------------------------------------- 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 [.9999890327, .9998788834, .9993031025, .9972193241, .9913586974, .9777250290, .9504576921, .9023388624, .8260254059, .7157948576, .5693457006, .3891005842, .\ 1825697217, -.3840385843e-1, -.2593774386, -.4656194467, -.6444699381, -.787249\ 7422, -.8903684896, -.9554961195, -.9888740299] The cut off is at j=, 14 --------------------------------------------- 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 [.9999942780, .9999339581, .9996035099, .9983518124, .9946712255, .9857273996, .9670552015, .9326746464, .8757950515, .7900806619, .6712233750, .5184068633, .\ 3352229679, .1297282134, -.8644159324e-1, -.2997291358, -.4967238800, -.6661972\ 996, -.8007000136, -.8974475798, -.9583985465, -.9895996366] The cut off is at j=, 15 --------------------------------------------- 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 [.9999970198, .9999641180, .9997756183, .9990298152, .9967415556, .9909446314, .9783363212, .9541624933, .9125297898, .8472235883, .7529377597, .6266518924, .\ 4687945583, .2838492793, .8018929939e-1, -.1308764979, -.3366656504, -.52496944\ 99, -.6858122788, -.8127934594, -.9037966389, -.9609986374, -.9902496593] The cut off is at j=, 16 --------------------------------------------- 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 [.9999984503, .9999805689, .9998736382, .9994325489, .9980229810, .9943068475, .9859013073, .9691502666, .9391747201, .8903256813, .8170521231, .7150519995, .\ 5824518389, .4207087858, .2349708841, .3375482398e-1, -.1720343284, -.370560334\ 3, -.5507043026, -.7035896440, -.8237138409, -.9095168387, -.9633387191, -.9908\ 346798] The cut off is at j=, 17 --------------------------------------------- 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 [.9999991953, .9999895096, .9999291599, .9996700585, .9988090862, .9964507706, .9909123024, .9794585994, .9581976631, .9222654374, .8663708642, .7856674697, .\ 6768019532, .5389056323, .3742743104, .1885364087, -.9777496724e-2, -.210212406\ 5, -.4017390980, -.5742251243, -.7197602090, -.8336141868, -.9146920195, -.9654\ 537930, -.9913634483] The cut off is at j=, 17 --------------------------------------------- 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 [.9999995828, .9999943525, .9999604523, .9998092055, .9992874041, .9978046184, .9941943577, .9864580845, .9715841785, .9455548429, .9036339130, .8409642400, .\ 7534110204, .6384974196, .4962234377, .3295596303, .1444666519, -.5060683019e-1 , -.2456803123, -.4304867691, -.5957858776, -.7345190580, -.8426228349, -.91939\ 21837, -.9673730267, -.9918432567] The cut off is at j=, 18 --------------------------------------------- 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 [.9999997839, .9999969676, .9999780077, .9998902306, .9995762585, .9986519245, .9963218328, .9911551075, .9808803698, .9622879872, .9313316703, .8834900896, .\ 8143855841, .7205785630, .6003883173, .4545574859, .2865916175, .1026651915, -.\ 8892483561e-1, -.2786814653, -.4570526972, -.6156049034, -.7480320301, -.850848\ 1223, -.9236761876, -.9691209003, -.9922802251] The cut off is at j=, 19 --------------------------------------------- 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 [.9999998882, .9999983758, .9999878146, .9999371460, .9997494845, .9991778387, .9976877486, .9942729590, .9872577932, .9741486451, .9516133952, .9156594283, .\ 8620438637, .7868904227, .6874226332, .5626734471, .4140140004, .2453667289, .\ 6302587553e-1, -.1249131619, -.3094351260, -.4816556258, -.6338707140, -.760440\ 8688, -.8583820601, -.9275938352, -.9707180951, -.9926795238] The cut off is at j=, 20 --------------------------------------------- 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 [.9999999423, .9999991320, .9999932712, .9999641692, .9998526983, .9995017713, .9985569678, .9963218328, .9915838121, .9824511055, .9662613076, .9396287396, .\ 8986811662, .8394978314, .7587078823, .6541561836, .5255085854, .3746629704, .2\ 058595440, .2543806804e-1, -.1587421887, -.3381385426, -.5044878890, -.65074673\ 46, -.7718673411, -.8653032376, -.9311875235, -.9721821904, -.9930455476] The cut off is at j=, 21 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 31], vs. those in the, 2, -th row from j=1 to j=, 30, as n goes to infinity, are as follows [.9999999702, .9999995371, .9999962961, .9999796590, .9999138275, .9996998752, .9991055634, .9976557037, .9944877602, .9881958726, .9767063385, .9572405991, .\ 9264198449, .8805443378, .8160426849, .7300404811, .6209568034, .4890136786, .3\ 365460676, .1680292345, -.1020972359e-1, -.1905705741, -.3649690824, -.52571901\ 17, -.6663752000, -.7824165552, -.8716791362, -.9344935451, -.9735282134, -.993\ 3820534] The cut off is at j=, 21 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 32], vs. those in the, 2, -th row from j=1 to j=, 31, as n goes to infinity, are as follows [.9999999846, .9999997537, .9999979673, .9999884975, .9999498291, .9998202234, .9994491499, .9985165579, .9964182258, .9921283024, .9840668213, .9700149906, .\ 9471275391, .9120837490, .8613954096, .7918545651, .7010651292, .5879710222, .4\ 532790823, .2996830104, .1318244462, -.4402738305e-1, -.2205453177, -.390086258\ 8, -.5454987882, -.6808803694, -.7921796500, -.8775679871, -.9375431285, -.9747\ 690784, -.9936922696] The cut off is at j=, 22 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 33], vs. those in the, 2, -th row from j=1 to j=, 32, as n goes to infinity, are as follows [.9999999921, .9999998691, .9999988875, .9999935193, .9999709209, .9998928811, .9996627983, .9990676350, .9976899422, .9947932548, .9891968547, .9791699712, .\ 9623858401, .9359772422, .8967236461, .8413760756, .7670938098, .6719348289, .5\ 553184307, .4183708841, .2640766483, .9718696465e-1, -.7612155298e-1, -.2488021\ 412, -.4136336118, -.5639599129, -.6943711913, -.8012359888, -.8830202726, -.94\ 03632762, -.9759159385, -.9939789846] The cut off is at j=, 23 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 34], vs. those in the, 2, -th row from j=1 to j=, 33, as n goes to infinity, are as follows [.9999999959, .9999999306, .9999993928, .9999963613, .9999832171, .9999364916, .9997947576, .9994177311, .9985206099, .9965823852, .9927357548, .9856579546, .\ 9734929857, .9538418820, .9238548082, .8804449489, .8206207366, .7419046678, .6\ 427807293, .5230954785, .3843353909, .2297170075, .6405445392e-1, -.1065945980, -.2754660555, -.4357404208, -.5812202292, -.7069435204, -.8096549592, -.8880799\ 494, -.9429774426, -.9769784706, -.9942446177] The cut off is at j=, 24 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 35], vs. those in the, 2, -th row from j=1 to j=, 34, as n goes to infinity, are as follows [.9999999979, .9999999633, .9999996694, .9999979637, .9999903525, .9999625244, .9998757485, .9996385609, .9990588826, .9977730090, .9951536368, .9902079690, .\ 9814871081, .9670361945, .9444173731, .9108318504, .8633523168, .7992549465, .7\ 164156551, .6137143113, .4913788871, .3512028802, .1965844968, .3236248715e-1, -.1355439907, -.3006520272, -.4565232505, -.5973846523, -.7186819705, -.8174974\ 582, -.8927854488, -.9454060874, -.9779651076, -.9944912769] The cut off is at j=, 25 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 36], vs. those in the, 2, -th row from j=1 to j=, 35, as n goes to infinity, are as follows [.9999999989, .9999999806, .9999998204, .9999988639, .9999944752, .9999779845, .9999251625, .9997769203, .9994050793, .9985588206, .9967907444, .9933677012, .\ 9871798923, .9766701369, .9598107376, .9341551300, .8969836531, .8455474917, .7\ 773945779, .6907403191, .5848295584, .4602286633, .3189907170, .1646524394, .\ 2046039760e-2, -.1630619968, -.3244656863, -.4760873341, -.6125468171, -.729661\ 4692, -.8248171240, -.8971705021, -.9476671305, -.9788832281, -.9947208070] The cut off is at j=, 26 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 37], vs. those in the, 2, -th row from j=1 to j=, 36, as n goes to infinity, are as follows [.9999999994, .9999999898, .9999999027, .9999993680, .9999968475, .9999871212, .9999551426, .9998630611, .9996261740, .9990734374, .9978898188, .9955417962, .\ 9911936062, .9836292993, .9712022236, .9518366974, .9231041504, .8823869046, .8\ 271277853, .7551455115, .6649782422, .5562050284, .4296907290, .2877059310, .13\ 38890665, -.2695942612e-1, -.1892355681, -.3470040394, -.4945278048, -.62679049\ 10, -.7399485670, -.8316613624, -.9012648233, -.9497763263, -.9797393134, -.994\ 9348284] The cut off is at j=, 26 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 38], vs. those in the, 2, -th row from j=1 to j=, 37, as n goes to infinity, are as follows [.9999999997, .9999999946, .9999999474, .9999996494, .9999982072, .9999924963, .9999732358, .9999163714, .9997664361, .9994079885, .9986217935, .9970247579, .\ 9939970445, .9886058170, .9795414454, .9650867943, .9431413213, .9113175792, .8\ 671179374, .8081850817, .7326036942, .6392161652, .5279057772, .3997991926, .25\ 73473393, .1042590810, -.5471718724e-1, -.2141463793, -.3683561633, -.511930789\ 8, -.6401907895, -.7496025455, -.8380722075, -.9050946788, -.9517475755, -.9805\ 390774, -.9951347693] The cut off is at j=, 27 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 39], vs. those in the, 2, -th row from j=1 to j=, 38, as n goes to infinity, are as follows [.9999999999, .9999999972, .9999999716, .9999998060, .9999989838, .9999956446, .9999841007, .9999491778, .9998548595, .9996239761, .9991056058, .9980280238, .\ 9959378864, .9921308503, .9855843598, .9749085444, .9583343411, .9337573974, .8\ 988510136, .8512513993, .7888052387, .7098554499, .6135291526, .4999851113, .37\ 05782996, .2279072896, .7572487901e-1, -.8128871925e-1, -.2378709615, -.3886038\ 671, -.5283743795, -.6528152229, -.7586763570, -.8440870447, -.9086833631, -.95\ 35931846, -.9812875744, -.9953218936] The cut off is at j=, 28 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 40], vs. those in the, 2, -th row from j=1 to j=, 39, as n goes to infinity, are as follows [.9999999999, .9999999985, .9999999847, .9999998929, .9999994257, .9999974812, .9999905941, .9999692584, .9999102715, .9997625061, .9994231076, .9987015126, .\ 9972703490, .9946054238, .9899214098, .9821147198, .9697291059, .9509612580, .9\ 237218120, .8857611197, .8348592822, .7690677915, .6869779543, .5879817433, .47\ 24861638, .3420440976, .1993730876, .4824749931e-1, -.1067333336, -.2604808973, -.4078223125, -.5439294849, -.6647246004, -.7672174257, -.8497392226, -.9120515\ 998, -.9553240839, -.9819892904, -.9954973226] The cut off is at j=, 29 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 41], vs. those in the, 2, -th row from j=1 to j=, 40, as n goes to infinity, are as follows [1.000000000, .9999999992, .9999999918, .9999999411, .9999996764, .9999985483, .9999944576, .9999814874, .9999448004, .9998508049, .9996300579, .9991503095, .\ 9981779161, .9963278754, .9930060044, .9873509143, .9781875739, .9640072010, .9\ 429887372, .9130743008, .8721045291, .8180102419, .7490458800, .6640398079, .56\ 26290552, .4454432964, .3142058518, .1717281629, .2178735711e-1, -.1311079474, -.2820430556, -.4260805917, -.5586605966, -.6759738130, -.7752683327, -.8550585\ 718, -.9152178791, -.9569500111, -.9826482188, -.9956620547] The cut off is at j=, 30 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 42], vs. those in the, 2, -th row from j=1 to j=, 41, as n goes to infinity, are as follows [1.000000000, .9999999996, .9999999956, .9999999676, .9999998182, .9999991661, .9999967464, .9999888990, .9999662012, .9999067546, .9997640829, .9994472966, .\ 9987914500, .9975171873, .9951810390, .9911209743, .9844055549, .9737984109, .9\ 577517058, .9344417554, .9018563871, .8579369777, .8007691514, .7288063390, .64\ 11016614, .5375178211, .4188833374, .2870672444, .1449530190, -.3695193640e-2, -.1544669522, -.3026198465, -.4434422590, -.5726264557, -.6866125115, -.7828674\ 031, -.8600718474, -.9181987457, -.9584796664, -.9832679252, -.9958169813] The cut off is at j=, 30 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 43], vs. those in the, 2, -th row from j=1 to j=, 42, as n goes to infinity, are as follows [1.000000000, .9999999998, .9999999976, .9999999822, .9999998981, .9999995225, .9999980969, .9999933702, .9999793976, .9999420078, .9998503495, .9996425254, .\ 9992032607, .9983321228, .9967022519, .9938119474, .9889345587, .9810754050, .9\ 689470815, .9509755575, .9253481642, .8901104985, .8433126500, .7831968368, .70\ 84099024, .6182168595, .5126873478, .3928266678, .2606273885, .1190260052, -.\ 2823943341e-1, -.1768621572, -.3222694855, -.4599658191, -.5858806482, -.696685\ 6979, -.7900492119, -.8648031066, -.9210090424, -.9599208442, -.9838516023, -.9\ 959629006] The cut off is at j=, 31 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 44], vs. those in the, 2, -th row from j=1 to j=, 43, as n goes to infinity, are as follows [1.000000000, .9999999999, .9999999987, .9999999903, .9999999431, .9999997274, .9999988907, .9999960559, .9999874953, .9999641018, .9999055528, .9997700538, .\ 9994778011, .9988864833, .9977579637, .9957169208, .9922046261, .9864339189, .9\ 773542148, .9636373130, .9436950480, .9157378918, .8778792427, .8282836967, .76\ 53499547, .6879114835, .5954320786, .4881703905, .3672881707, .2348816824, .\ 9392394180e-1, -.5188395764e-1, -.1983427852, -.3410462582, -.4757051727, -.598\ 4721311, -.7062342391, -.7968450205, -.8692740333, -.9236621198, -.9612805464, -.9844021158, -.9961005290] The cut off is at j=, 32 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 45], vs. those in the, 2, -th row from j=1 to j=, 44, as n goes to infinity, are as follows [1.000000000, .9999999999, .9999999993, .9999999947, .9999999682, .9999998448, .9999993555, .9999976623, .9999924414, .9999778780, .9999406822, .9998528588, .\ 9996596472, .9992609846, .9984852319, .9970549378, .9945461559, .9903451760, .9\ 836091221, .9732391443, .9578762142, .9359291713, .9056422520, .8652048254, .81\ 28998931, .7472809781, .6673605047, .5727879444, .4639939466, .3422780562, .209\ 8225285, .6962262523e-1, -.7466644193e-1, -.2189555091, -.3590007802, -.4907100\ 232, -.6104456987, -.7152953172, -.8032831550, -.8735042178, -.9261700149, -.96\ 25650780, -.9849220453, -.9962305113] The cut off is at j=, 33 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 46], vs. those in the, 2, -th row from j=1 to j=, 45, as n goes to infinity, are as follows [1.000000000, 1.000000000, .9999999996, .9999999971, .9999999823, .9999999119, .9999996267, .9999986193, .9999954491, .9999864261, .9999629188, .9999063165, .\ 9997793544, .9995123154, .9989827254, .9979877383, .9962095874, .9931762712, .9\ 882218547, .9804530419, .9687304583, .9516737890, .9276990302, .8950933581, .85\ 21285923, .7972084134, .7290382613, .6468012524, .5503196188, .4401799680, .317\ 8025783, .1854399367, .4609723384e-1, -.9662347389e-1, -.2387445148, -.37618024\ 66, -.5050262453, -.6218423963, -.7239028231, -.8093893344, -.8775113980, -.928\ 5436060, -.9637801306, -.9854137177, -.9963534294] The cut off is at j=, 34 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 47], vs. those in the, 2, -th row from j=1 to j=, 46, as n goes to infinity, are as follows [1.000000000, 1.000000000, .9999999998, .9999999984, .9999999902, .9999999501, .9999997845, .9999991873, .9999972703, .9999917055, .9999769231, .9999406392, .\ 9998576927, .9996799250, .9993207616, .9986334214, .9973824103, .9952092384, .9\ 915950744, .9858250933, .9769612085, .9638312022, .9450424728, .9190273089, .88\ 41236307, .8386906762, .7812536970, .7106661636, .6262732453, .5280573490, .416\ 7459998, .2938646538, .1617220298, .2332265282e-1, -.1177904374, -.2577515841, -.3926286720, -.5186962204, -.6326998885, -.7320877018, -.8151869568, -.8813116\ 688, -.9307927459, -.9649308533, -.9858792374, -.9964698093] The cut off is at j=, 35 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 48], vs. those in the, 2, -th row from j=1 to j=, 47, as n goes to infinity, are as follows [1.000000000, 1.000000000, .9999999999, .9999999991, .9999999946, .9999999718, .9999998759, .9999995232, .9999983686, .9999949516, .9999857002, .9999625607, .\ 9999086702, .9997910298, .9995489800, .9990773286, .9982033864, .9966580197, .9\ 940421646, .9897919553, .9831474614, .9731316233, .9585468721, .9379966992, .90\ 99378094, .8727653921, .8249297333, .7650773873, .6922052127, .6058116002, .506\ 0269778, .3937057474, .2704643974, .1386554670, .1273732470e-2, -.1382014380, -\ .2760161897, -.4083871156, -.5317591407, -.6430527858, -.7398782570, -.82069734\ 95, -.8849196640, -.9329263776, -.9660219150, -.9863205112, -.9965801278] The cut off is at j=, 36 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 49], vs. those in the, 2, -th row from j=1 to j=, 48, as n goes to infinity, are as follows [1.000000000, 1.000000000, .9999999999, .9999999995, .9999999970, .9999999841, .9999999288, .9999997212, .9999990284, .9999969390, .9999911755, .9999764916, .\ 9999416644, .9998642550, .9997021135, .9993805331, .9987741242, .9976830078, .9\ 958038629, .9926977107, .9877579266, .9801835910, .9689645515, .9528850898, .93\ 05525039, .9004550498, .8610505368, .8108827541, .7487183277, .6736922957, .585\ 4473915, .4842504127, .3710695811, .2475995830, .1162257960, -.2007450791e-1, -\ .1578892597, -.2935755983, -.4234938941, -.5442512845, -.6529329359, -.74730041\ 85, -.8259399874, -.8883487150, -.9349526350, -.9670575576, -.9867392711, -.996\ 6848178] The cut off is at j=, 36 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 50], vs. those in the, 2, -th row from j=1 to j=, 49, as n goes to infinity, are as follows [1.000000000, 1.000000000, 1.000000000, .9999999997, .9999999983, .9999999911, .9999999592, .9999998374, .9999994233, .9999981509, .9999945760, .9999853020, .\ 9999629090, .9999122488, .9998042626, .9995863446, .9991682899, .9984031446, .9\ 970628900, .9948098942, .9911663776, .9854856259, .9769300695, .9644623022, .94\ 68552916, .9227271659, .8906039216, .8490102690, .7965849360, .7322126006, .655\ 1608659, .5652079958, .4627460547, .3488449814, .2252660392, .9441774740e-1, -.\ 4074672067e-1, -.1768853471, -.3104649767, -.4379847810, -.5562062662, -.662369\ 6842, -.7543779797, -.8309326856, -.8916109894, -.9368789303, -.9680416433, -.9\ 871370929, -.9967842732] The cut off is at j=, 37 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 51], vs. those in the, 2, -th row from j=1 to j=, 50, as n goes to infinity, are as follows [1.000000000, 1.000000000, 1.000000000, .9999999999, .9999999991, .9999999950, .9999999767, .9999999055, .9999996588, .9999988869, .9999966789, .9999908482, .\ 9999765204, .9999435389, .9998720187, .9997252139, .9994387959, .9989057401, .9\ 979563931, .9963339796, .9936668302, .9894398746, .9829692858, .9733853122, .95\ 96290052, .9404684348, .9145388879, .8804093989, .8366748965, .7820696055, .715\ 5935990, .6366411573, .5451174191, .4415291881, .3270369328, .2034579906, .\ 7321547790e-1, -.6076710690e-1, -.1952198060, -.3267175008, -.4518931910, -.567\ 6552622, -.6713901053, -.7611328067, -.8356917683, -.8947176128, -.9387120311, -.9689776948, -.9875154137, -.9968788534] The cut off is at j=, 38 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 52], vs. those in the, 2, -th row from j=1 to j=, 51, as n goes to infinity, are as follows [1.000000000, 1.000000000, 1.000000000, .9999999999, .9999999995, .9999999972, .9999999867, .9999999452, .9999997987, .9999993322, .9999979739, .9999943242, .\ 9999851997, .9999638352, .9999167188, .9998183785, .9996233152, .9992542766, .9\ 985862483, .9974259887, .9954876726, .9923662285, .9875111414, .9802046780, .96\ 95494189, .9544703901, .9337367255, .9060065157, .8698962974, .8240736755, .767\ 3681810, .6988921210, .6181604025, .5251966054, .4206123338, .3056482697, .1821\ 683491, .5260277255e-1, -.8015932313e-1, -.2129214188, -.3423644621, -.46525035\ 31, -.5786272168, -.6800192112, -.7675850245, -.8402322178, -.8976787756, -.940\ 4581271, -.9698689312, -.9878755460, -.9969688865] The cut off is at j=, 39 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 53], vs. those in the, 2, -th row from j=1 to j=, 52, as n goes to infinity, are as follows [1.000000000, 1.000000000, 1.000000000, 1.000000000, .9999999997, .9999999984, .9999999925, .9999999683, .9999998816, .9999996007, .9999987684, .9999964934, .\ 9999907088, .9999769361, .9999460555, .9998805370, .9997484501, .9994944998, .9\ 990273990, .9982031437, .9968042646, .9945159022, .9909005418, .9853743455, .97\ 71890386, .9654240242, .9489935730, .9266733702, .8971492923, .8590890752, .811\ 2346956, .7525101680, .6821364813, .5997430465, .5054637240, .4000055676, .2846\ 799813, .1613889641, .3256321094e-1, -.9894641214e-1, -.2300176698, -.357435373\ 2, -.4780854697, -.5891490275, -.6882801370, -.7737531825, -.8445678071, -.9005\ 038271, -.9421228896, -.9707182999, -.9882186910, -.9970546727] The cut off is at j=, 40 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 54], vs. those in the, 2, -th row from j=1 to j=, 53, as n goes to infinity, are as follows [1.000000000, 1.000000000, 1.000000000, 1.000000000, .9999999998, .9999999991, .9999999957, .9999999817, .9999999306, .9999997620, .9999992539, .9999978415, .\ 9999941903, .9999853528, .9999652129, .9999217894, .9998328390, .9996590972, .9\ 993344746, .9987526473, .9977498033, .9960838542, .9934112088, .9892631503, .98\ 30248593, .9739209791, .9610121438, .9432068538, .9192923507, .8879866475, .848\ 0116725, .7981848006, .7375231791, .6653526334, .5814109560, .4859344358, .3797\ 168070, .2641314784, .1411108363, .1308030381e-1, -.1171507535, -.2465347795, -\ .3719580700, -.4904258650, -.5992457146, -.6961943079, -.7796544012, -.84871121\ 60, -.9032013589, -.9437115235, -.9715285032, -.9885459495, -.9971364874] The cut off is at j=, 41 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 55], vs. those in the, 2, -th row from j=1 to j=, 54, as n goes to infinity, are as follows [1.000000000, 1.000000000, 1.000000000, 1.000000000, .9999999999, .9999999995, .9999999976, .9999999895, .9999999594, .9999998585, .9999995495, .9999986761, .\ 9999963810, .9999907356, .9999776630, .9999490278, .9998894438, .9997712396, .9\ 995469549, .9991387690, .9984244438, .9972197286, .9952577638, .9921667912, .98\ 74483747, .9804592203, .9704003876, .9563180219, .9371195257, .9116082235, .878\ 5380170, .8366873835, .7849495377, .7224329741, .6485642980, .5631836203, .4666\ 221396, .3597520684, .2440008279, .1213242995, -.5862395429e-2, -.1347940298, -\ .2624977438, -.3859588094, -.5022971214, -.6089405739, -.7037815891, -.78530450\ 42, -.8526741355, -.9057792785, -.9452288133, -.9723020234, -.9888583327, -.997\ 2145832] The cut off is at j=, 41 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 56], vs. those in the, 2, -th row from j=1 to j=, 55, as n goes to infinity, are as follows [1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, .9999999997, .9999999986, .9999999940, .9999999763, .9999999162, .9999997289, .9999991908, .\ 9999977540, .9999941632, .9999857167, .9999669248, .9999272158, .9998472280, .9\ 996931405, .9994084632, .9989027866, .9980372139, .9966066146, .9943194329, .99\ 07765435, .9854514735, .9776750791, .9666283254, .9513469827, .9307417002, .903\ 6359416, .8688226914, .8251387572, .7715531312, .7072635138, .6317930934, .5450\ 783427, .4475381983, .3401156973, .2242849573, .1020191761, -.2428120033e-1, -.\ 1518972057, -.2779303770, -.3994623635, -.5137232056, -.6182553165, -.711060420\ 8, -.7907181353, -.8564673600, -.9082448744, -.9466791638, -.9730411444, -.9891\ 567703, -.9972891926] The cut off is at j=, 42 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 57], vs. those in the, 2, -th row from j=1 to j=, 56, as n goes to infinity, are as follows [1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, .9999999998, .9999999992, .9999999965, .9999999862, .9999999505, .9999998373, .9999995070, .\ 9999986110, .9999963366, .9999909031, .9999786285, .9999522951, .9998984482, .9\ 997931622, .9995957509, .9992399043, .9986218551, .9975854457, .9959044097, .99\ 32627816, .9892350759, .9832686259, .9746711373, .9626069194, .9461052880, .924\ 0841454, .8953906981, .8588596888, .8133875234, .7580184749, .6920370255, .6150\ 586678, .5271104214, .4286921455, .3208105739, .2049798339, .8318490948e-1, -.\ 4219221864e-1, -.1684805177, -.2928553578, -.4124921087, -.5247265857, -.627210\ 1950, -.7180479396, -.7959088636, -.8601008698, -.9106048746, -.9480666365, -.9\ 737479701, -.9894421184, -.9973605296] The cut off is at j=, 43 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 58], vs. those in the, 2, -th row from j=1 to j=, 57, as n goes to infinity, are as follows [1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, .9999999999, .9999999996, .9999999980, .9999999920, .9999999708, .9999999027, .9999997006, .\ 9999991440, .9999977091, .9999942288, .9999862472, .9999688670, .9999327989, .9\ 998612351, .9997250893, .9994761128, .9990374399, .9982912762, .9970637448, .99\ 51073667, .9920822472, .9875377237, .9808969049, .9714470791, .9583392561, .940\ 6000024, .9171581409, .8868877908, .8486676518, .8014545388, .7443671406, .6767\ 740753, .5983788280, .5092933197, .4100918746, .3018382976, .1860806215, .\ 6481067519e-1, -.5961129437e-1, -.1845634712, -.3072942760, -.4250701104, -.535\ 3283383, -.6358241190, -.7247600882, -.8008892778, -.8635839046, -.9128654982, -.9493949813, -.9744244420, -.9897151671, -.9974287918] The cut off is at j=, 44 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 59], vs. those in the, 2, -th row from j=1 to j=, 58, as n goes to infinity, are as follows [1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, .9999999998, .9999999989, .9999999954, .9999999829, .9999999419, .9999998188, .\ 9999994742, .9999985725, .9999963523, .9999911850, .9999797665, .9999557235, .9\ 999073270, .9998139302, .9996406905, .9993311277, .9987971499, .9979063723, .99\ 64668837, .9942100855, .9907728082, .9856805456, .9783342403, .9680034985, .953\ 8292763, .9348388669, .9099753460, .8781425047, .8382647625, .7893597511, .7306\ 193985, .6614936457, .5817696654, .4916388252, .3917438107, .2831993497, .16758\ 18161, .4688547618e-1, -.7655396242e-1, -.2001648447, -.3212676801, -.437217203\ 3, -.5455482482, -.6441147602, -.7312117145, -.8056710714, -.8669250296, -.9150\ 325014, -.9506676658, -.9750723541, -.9899766459, -.9974941615] The cut off is at j=, 45 --------------------------------------------- The limits of the sorting probabilities of the cell, [1, 60], vs. those in the, 2, -th row from j=1 to j=, 59, as n goes to infinity, are as follows [1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, .9999999999, .9999999994, .9999999973, .9999999900, .9999999655, .9999998906, .\ 9999996781, .9999991135, .9999977029, .9999943715, .9999869030, .9999709507, .9\ 999383814, .9998746381, .9997547400, .9995375029, .9991575849, .9985150703, .99\ 74625300, .9957898443, .9932075463, .9893300005, .9836603116, .9755793758, .964\ 3418244, .9490816740, .9288301827, .9025476830, .8691700127, .8276686767, .7771\ 221776, .7167942479, .6462132162, .5652456781, .4741571977, .3736530672, .26489\ 32402, .1494773652, .2939822238e-1, -.9303541339e-1, -.2153026991, -.3347951244 , -.4489530665, -.5554048994, -.6520986477, -.7374166609, -.8102651183, -.87013\ 21944, -.9171112194, -.9518879002, -.9756933662, -.9902272297, -.9975568074] The cut off is at j=, 46 ------------------------- This ends this article that took, 732.681, seconds to produce ----------------------- This took, 732.681, seconds.