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]]}