The Minimal sorting probabilities, and the champion pairs for Standard Young\ Tableau of shape, [n, n, n], for n from 1 to, 32 By Shalosh B. Ekhad The minimal sorting probability of Standard Young tableaux of shape, [2, 2, 2], but where one of the cells is in the first row is , 1/5, and in floating point, 0.2000000000, and this happens with the pairs of cells, {[[1, 2], [2, 1]]} The minimal sorting probability of Standard Young tableaux of shape, [3, 3, 3], but where one of the cells is in the first row is , 0, and in floating point, 0., and this happens with the pairs of cells, {[[1, 2], [2, 1]], [[1, 3], [2, 2]], [[1, 3], [3, 1]]} The minimal sorting probability of Standard Young tableaux of shape, [4, 4, 4], but where one of the cells is in the first row is , 1/11, and in floating point, 0.09090909091, and this happens with the pairs of cells, {[[1, 2], [2, 1]]} The minimal sorting probability of Standard Young tableaux of shape, [5, 5, 5], but where one of the cells is in the first row is , 1/7, and in floating point, 0.1428571429, and this happens with the pairs of cells, {[[1, 2], [2, 1]]} The minimal sorting probability of Standard Young tableaux of shape, [6, 6, 6], 18 but where one of the cells is in the first row is , ---, 221 and in floating point, 0.08144796380, and this happens with the pairs of cells, {[[1, 4], [3, 1]], [[1, 6], [3, 3]]} The minimal sorting probability of Standard Young tableaux of shape, [7, 7, 7], but where one of the cells is in the first row is , 4/323, and in floating point, 0.01238390093, and this happens with the pairs of cells, {[[1, 4], [3, 1]], [[1, 7], [3, 4]]} The minimal sorting probability of Standard Young tableaux of shape, [8, 8, 8], 6827 but where one of the cells is in the first row is , -----, 96577 and in floating point, 0.07068970873, and this happens with the pairs of cells, {[[1, 6], [2, 4]]} The minimal sorting probability of Standard Young tableaux of shape, [9, 9, 9], 371 but where one of the cells is in the first row is , ------, 482885 and in floating point, 0.0007682988703, and this happens with the pairs of cells, {[[1, 7], [2, 5]]} The minimal sorting probability of Standard Young tableaux of shape, 1423 [10, 10, 10], but where one of the cells is in the first row is , -----, 43355 and in floating point, 0.03282205051, and this happens with the pairs of cells, {[[1, 5], [2, 3]]} The minimal sorting probability of Standard Young tableaux of shape, 284645 [11, 11, 11], but where one of the cells is in the first row is , -------, 5500082 and in floating point, 0.05175286478, and this happens with the pairs of cells, {[[1, 10], [2, 8]]} The minimal sorting probability of Standard Young tableaux of shape, 47569 [12, 12, 12], but where one of the cells is in the first row is , -------, 1696940 and in floating point, 0.02803222271, and this happens with the pairs of cells, {[[1, 11], [2, 9]]} The minimal sorting probability of Standard Young tableaux of shape, 101887 [13, 13, 13], but where one of the cells is in the first row is , --------, 12369285 and in floating point, 0.008237096970, and this happens with the pairs of cells, {[[1, 12], [2, 10]]} The minimal sorting probability of Standard Young tableaux of shape, 106957 [14, 14, 14], but where one of the cells is in the first row is , --------, 12538005 and in floating point, 0.008530623492, and this happens with the pairs of cells, {[[1, 13], [2, 11]]} The minimal sorting probability of Standard Young tableaux of shape, [15, 15, 15], but where one of the cells is in the first row is , 3595555 ---------, and in floating point, 0.02291494070, 156908763 and this happens with the pairs of cells, {[[1, 14], [2, 12]]} The minimal sorting probability of Standard Young tableaux of shape, 26797 [16, 16, 16], but where one of the cells is in the first row is , -------, 1533939 and in floating point, 0.01746940393, and this happens with the pairs of cells, {[[1, 4], [2, 2]]} The minimal sorting probability of Standard Young tableaux of shape, [17, 17, 17], but where one of the cells is in the first row is , 1672986859 --------------, and in floating point, 0.00003968546399, 42156162250685 and this happens with the pairs of cells, {[[1, 11], [2, 8]]} The minimal sorting probability of Standard Young tableaux of shape, 29 [18, 18, 18], but where one of the cells is in the first row is , ------, 168805 and in floating point, 0.0001717958591, and this happens with the pairs of cells, {[[1, 4], [2, 2]]} The minimal sorting probability of Standard Young tableaux of shape, 30763 [19, 19, 19], but where one of the cells is in the first row is , --------, 22057805 and in floating point, 0.001394653729, and this happens with the pairs of cells, {[[1, 7], [3, 2]], [[1, 18], [3, 13]]} The minimal sorting probability of Standard Young tableaux of shape, 23 [20, 20, 20], but where one of the cells is in the first row is , ----, 1711 and in floating point, 0.01344243133, and this happens with the pairs of cells, {[[1, 4], [2, 2]]} The minimal sorting probability of Standard Young tableaux of shape, [21, 21, 21], but where one of the cells is in the first row is , 14896330 ----------, and in floating point, 0.006682185450, 2229260189 and this happens with the pairs of cells, {[[1, 8], [2, 5]]} The minimal sorting probability of Standard Young tableaux of shape, [22, 22, 22], but where one of the cells is in the first row is , 628214285 -----------, and in floating point, 0.007827886979, 80253366804 and this happens with the pairs of cells, {[[1, 8], [2, 5]]} The minimal sorting probability of Standard Young tableaux of shape, [23, 23, 23], but where one of the cells is in the first row is , 7920217337 ------------, and in floating point, 0.008837924471, 896162595978 and this happens with the pairs of cells, {[[1, 9], [3, 3]], [[1, 21], [3, 15]]} The minimal sorting probability of Standard Young tableaux of shape, [24, 24, 24], but where one of the cells is in the first row is , 929503417 -----------, and in floating point, 0.01006526122, 92347669542 and this happens with the pairs of cells, {[[1, 9], [3, 3]], [[1, 22], [3, 16]]} The minimal sorting probability of Standard Young tableaux of shape, [25, 25, 25], but where one of the cells is in the first row is , 6084420897762 ---------------, and in floating point, 0.008188538066, 743041169057383 and this happens with the pairs of cells, {[[1, 22], [2, 19]]} The minimal sorting probability of Standard Young tableaux of shape, [26, 26, 26], but where one of the cells is in the first row is , 3770189291461 ----------------, and in floating point, 0.001211609513, 3111719783961901 and this happens with the pairs of cells, {[[1, 23], [2, 20]]} The minimal sorting probability of Standard Young tableaux of shape, [27, 27, 27], but where one of the cells is in the first row is , 888263684500761353 ---------------------, and in floating point, 0.001977384667, 449211374718626598428 and this happens with the pairs of cells, {[[1, 17], [3, 9]], [[1, 19], [3, 11]]} The minimal sorting probability of Standard Young tableaux of shape, [28, 28, 28], but where one of the cells is in the first row is , 61606 ---------, and in floating point, 0.0003654823682, 168560799 and this happens with the pairs of cells, {[[1, 5], [3, 1]], [[1, 28], [3, 24]]} The minimal sorting probability of Standard Young tableaux of shape, [29, 29, 29], but where one of the cells is in the first row is , 504413453295858169 ---------------------, and in floating point, 0.004450960314, 113326881767920995956 and this happens with the pairs of cells, {[[1, 15], [3, 7]], [[1, 23], [3, 15]]} The minimal sorting probability of Standard Young tableaux of shape, [30, 30, 30], but where one of the cells is in the first row is , 34380314273872238772784 -------------------------, and in floating point, 0.003767543755, 9125392168877860377115999 and this happens with the pairs of cells, {[[1, 22], [2, 18]]} The minimal sorting probability of Standard Young tableaux of shape, [31, 31, 31], but where one of the cells is in the first row is , 6417302959421 -----------------, and in floating point, 0.0005706203444, 11246186755791453 and this happens with the pairs of cells, {[[1, 11], [3, 4]], [[1, 28], [3, 21]]} The minimal sorting probability of Standard Young tableaux of shape, [32, 32, 32], but where one of the cells is in the first row is , 4956330014850197 -------------------, and in floating point, 0.001339550303, 3699995442655388037 and this happens with the pairs of cells, {[[1, 13], [2, 9]]}