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]]}