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, n], as n goes to infinity for i from 2 to, 16 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 [-.2500000000] 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 [-.8750000000] The cut off is at j=, 1 The limits of the sorting probabilities of the cell, [1, 2], vs. those in the, 4, -th row from j=1 to j=, 1, as n goes to infinity, are as follows [-.9921875000] 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 [.3750000000, -.5585937500] 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 [-.5625000000, -.9711914062] The cut off is at j=, 1 The limits of the sorting probabilities of the cell, [1, 3], vs. those in the, 4, -th row from j=1 to j=, 2, as n goes to infinity, are as follows [-.9516601562, -.9995727539] 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 [.7265625000, .4882812500e-2, -.6988983154] The cut off is at j=, 3 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 [-.1586914062, -.8474731445, -.9899063110] The cut off is at j=, 1 The limits of the sorting probabilities of the cell, [1, 4], vs. those in the, 4, -th row from j=1 to j=, 3, as n goes to infinity, are as follows [-.8536605835, -.9951133728, -.9999449253] 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 [.8906250000, .4493408203, -.2279281616, -.7775691748] 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 [.2215576172, -.6031265259, -.9331836700, -.9955733716] The cut off is at j=, 2 The limits of the sorting probabilities of the cell, [1, 5], vs. those in the, 4, -th row from j=1 to j=, 4, as n goes to infinity, are as follows [-.6916699409, -.9763718843, -.9991178215, -.9999888130] 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 [.9589843750, .7256774902, .2210413218, -.3842742220, -.8270631481] 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 [.5174484253, -.2784743309, -.7912320048, -.9662123220, -.9977604431] The cut off is at j=, 2 The limits of the sorting probabilities of the cell, [1, 6], vs. those in the, 4, -th row from j=1 to j=, 5, as n goes to infinity, are as follows [-.4790973356, -.9281271277, -.9943753127, -.9997775576, -.9999969754] The cut off is at j=, 1 --------------------------------------------- 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 [.9853515625, .8738403320, .5541627407, .3988571186e-1, -.4948451448, -.8606198\ 697] The cut off is at j=, 5 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 [.7198128700, .5882309377e-1, -.5587753663, -.8804509981, -.9811187809, -.99874\ 80598] The cut off is at j=, 3 The limits of the sorting probabilities of the cell, [1, 7], vs. those in the, 4, -th row from j=1 to j=, 6, as n goes to infinity, are as follows [-.2403236853, -.8371775120, -.9784422539, -.9982919769, -.9999300241, -.999999\ 0036] The cut off is at j=, 1 --------------------------------------------- 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 [.9949645996, .9455108643, .7649301291, .3952118484, -.1038218591, -.5762752719 , -.8846134596] 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 [.8458195478, .3547862666, -.2676476412, -.7184207032, -.9268642606, -.98863903\ 21, -.9992464762] The cut off is at j=, 3 The limits of the sorting probabilities of the cell, [1, 8], vs. those in the, 4, -th row from j=1 to j=, 7, as n goes to infinity, are as follows [-.5484390722e-3, -.6990094341, -.9406232287, -.9923054546, -.9993853421, -.999\ 9742080, -.9999996205] The cut off is at j=, 1 --------------------------------------------- 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 [.9983215332, .9776277542, .8840546012, .6482240148, .2542385573, -.2189692421, -.6382109932, -.9024689377] 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 [.9188619582, .5842477479, .3482973675e-1, -.4867139515, -.8130337112, -.952848\ 4682, -.9927625355, -.9995198421] The cut off is at j=, 4 The limits of the sorting probabilities of the cell, [1, 9], vs. those in the, 4, -th row from j=1 to j=, 8, as n goes to infinity, are as follows [.2206926201, -.5198134332, -.8703524856, -.9755835807, -.9968771832, -.9997490\ 738, -.9999892867, -.9999998385] The cut off is at j=, 2 --------------------------------------------- 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 [.9994544983, .9911897182, .9458826510, .8085669489, .5329292834, .1313379231, -.3124107033, -.6865705148, -.9161785177] The cut off is at j=, 7 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 [.9588754029, .7460512025, .3084843887, -.2143026350, -.6331855128, -.871466556\ 0, -.9682807378, -.9951750473, -.9996798178] The cut off is at j=, 4 The limits of the sorting probabilities of the cell, [1, 10], vs. those in the, 4, -th row from j=1 to j=, 9, as n goes to infinity, are as follows [.4117518240, -.3138098155, -.7614931222, -.9396091427, -.9889600700, -.9986004\ 482, -.9998870443, -.9999951136, -.9999999249] The cut off is at j=, 2 --------------------------------------------- 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 [.9998264313, .9966498017, .9758871710, .9015257411, .7258112301, .4236607812, .2477818705e-1, -.3892176457, -.7251575098, -.9269731482] The cut off is at j=, 8 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 [.9798126205, .8518850313, .5309389545, .6239045372e-1, -.4006901290, -.7324564\ 456, -.9089434955, -.9779001576, -.9966619083, -.9997784354] The cut off is at j=, 5 The limits of the sorting probabilities of the cell, [1, 11], vs. those in the, 4, -th row from j=1 to j=, 10, as n goes to infinity, are as follows [.5681910983, -.9840226154e-1, -.6149074674, -.8759714370, -.9699013685, -.9946\ 107397, -.9993211247, -.9999450195, -.9999975970, -.9999999625] The cut off is at j=, 2 --------------------------------------------- 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 [.9999457598, .9987634569, .9896737253, .9517270203, .8477618927, .6406655847, .3223862231, -.6761778122e-1, -.4531206652, -.7565130402, -.9356506270] 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 [.9903596572, .9170060978, .6969672159, .3131828519, -.1421731714, -.5386711261 , -.8010179253, -.9337971325, -.9841424105, -.9976182719, -.9998418809] The cut off is at j=, 5 The limits of the sorting probabilities of the cell, [1, 12], vs. those in the, 4, -th row from j=1 to j=, 11, as n goes to infinity, are as follows [.6907994878, .1102961727, -.4382755640, -.7792750017, -.9325033495, -.98410844\ 92, -.9971978180, -.9996487005, -.9999714700, -.9999987427, -.9999999801] The cut off is at j=, 3 --------------------------------------------- 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 [.9999833107, .9995551761, .9957267039, .9773029577, .9194640691, .7878064993, .5563151117, .2296480835, -.1479527923, -.5068822512, -.7823908869, -.942748503\ 9] 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 [.9955052658, .9550944371, .8125165888, .5206279142, .1133014744, -.3068925964, -.6410153284, -.8493204250, -.9507701016, -.9883331176, -.9982557866, -.9998841\ 823] The cut off is at j=, 6 The limits of the sorting probabilities of the cell, [1, 13], vs. those in the, 4, -th row from j=1 to j=, 12, as n goes to infinity, are as follows [.7834391532, .3002259006, -.2437651045, -.6491369561, -.8699521074, -.96170233\ 24, -.9911853924, -.9984639082, -.9998081435, -.9999843833, -.9999993070, -.999\ 9999889] The cut off is at j=, 3 --------------------------------------------- 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 [.9999949336, .9998435271, .9982837121, .9897106384, .9591634568, .8805369681, .7243522816, .4746969160, .1452612016, -.2180842274, -.5525710127, -.8040355482 , -.9486408588] The cut off is at j=, 10 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 [.9979479089, .9764382976, .8883779561, .6797769295, .3428475666, -.6110432680e\ -1, -.4377748901, -.7174980048, -.8840118658, -.9626636396, -.9912294588, -.998\ 6938433, -.9999132545] The cut off is at j=, 6 The limits of the sorting probabilities of the cell, [1, 14], vs. those in the, 4, -th row from j=1 to j=, 13, as n goes to infinity, are as follows [.8512931729, .4642835271, -.4490515349e-1, -.4904932478, -.7779163649, -.92116\ 80405, -.9774405728, -.9948986151, -.9991192684, -.9998903262, -.9999910557, -.\ 9999996008, -.9999999936] The cut off is at j=, 3 --------------------------------------------- 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 [.9999984801, .9999460326, .9993286257, .9954833012, .9800546213, .9355262151, .8365552178, .6594635225, .3968992584, .6869646933e-1, -.2795913461, -.59175567\ 93, -.8223515177, -.9535952137] The cut off is at j=, 11 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 [.9990803286, .9879687072, .9357791411, .7944043181, .5334943988, .1753317973, -.2092074774, -.5413707474, -.7752229258, -.9093811496, -.9711891269, -.9932821\ 397, -.9990028010, -.9999337625] The cut off is at j=, 7 The limits of the sorting probabilities of the cell, [1, 15], vs. those in the, 4, -th row from j=1 to j=, 14, as n goes to infinity, are as follows [.8996823845, .5998754114, .1461326752, -.3123431929, -.6559997749, -.856963984\ 2, -.9508925088, -.9862601871, -.9969366580, -.9994751306, -.9999348002, -.9999\ 946741, -.9999997612, -.9999999961] The cut off is at j=, 4 --------------------------------------------- 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 [.9999995488, .9999817093, .9997434561, .9980733215, .9905778516, .9664901016, .9070410405, .7890285383, .5946386250, .3234573220, -.7128181804e-3, -.33379428\ 50, -.6256412029, -.8380100093, -.9578074130] 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 [.9995945836, .9940031886, .9641642223, .8726417593, .6816843225, .3851090368, .2363839666e-1, -.3331875501, -.6234672325, -.8192626789, -.9282446230, -.97742\ 43667, -.9947691641, -.9992257041, -.9999485608] The cut off is at j=, 8 The limits of the sorting probabilities of the cell, [1, 16], vs. those in the, 4, -th row from j=1 to j=, 15, as n goes to infinity, are as follows [.9333983202, .7077887366, .3202154832, -.1256458195, -.5080361623, -.765799433\ 6, -.9061462663, -.9686214140, -.9913790958, -.9981000990, -.9996765176, -.9999\ 598988, -.9999967197, -.9999998523, -.9999999976] The cut off is at j=, 4 ------------------------- This ends this article that took, 7009.257, seconds to produce ----------------------- This took, 7009.257, seconds.