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Statistic identifier: St000810
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions.
For example, $p_{22} = 2m_{22} + m_4$, so the statistic on the partition $22$ is 3.
-----------------------------------------------------------------------------
References:
-----------------------------------------------------------------------------
Code:
def statistic(mu):
m = SymmetricFunctions(ZZ).m()
p = SymmetricFunctions(ZZ).p()
return sum(coeff for _, coeff in m(p(mu)))
-----------------------------------------------------------------------------
Statistic values:
[] => 1
[1] => 1
[2] => 1
[1,1] => 3
[3] => 1
[2,1] => 2
[1,1,1] => 10
[4] => 1
[3,1] => 2
[2,2] => 3
[2,1,1] => 7
[1,1,1,1] => 47
[5] => 1
[4,1] => 2
[3,2] => 2
[3,1,1] => 6
[2,2,1] => 6
[2,1,1,1] => 26
[1,1,1,1,1] => 246
[6] => 1
[5,1] => 2
[4,2] => 2
[4,1,1] => 6
[3,3] => 3
[3,2,1] => 6
[3,1,1,1] => 24
[2,2,2] => 10
[2,2,1,1] => 26
[2,1,1,1,1] => 138
[1,1,1,1,1,1] => 1602
[7] => 1
[6,1] => 2
[5,2] => 2
[5,1,1] => 6
[4,3] => 2
[4,2,1] => 5
[4,1,1,1] => 23
[3,3,1] => 6
[3,2,2] => 6
[3,2,1,1] => 20
[3,1,1,1,1] => 114
[2,2,2,1] => 23
[2,2,1,1,1] => 105
[2,1,1,1,1,1] => 767
[1,1,1,1,1,1,1] => 11481
[8] => 1
[7,1] => 2
[6,2] => 2
[6,1,1] => 6
[5,3] => 2
[5,2,1] => 5
[5,1,1,1] => 23
[4,4] => 3
[4,3,1] => 6
[4,2,2] => 7
[4,2,1,1] => 19
[4,1,1,1,1] => 111
[3,3,2] => 6
[3,3,1,1] => 22
[3,2,2,1] => 20
[3,2,1,1,1] => 90
[3,1,1,1,1,1] => 652
[2,2,2,2] => 47
[2,2,2,1,1] => 111
[2,2,1,1,1,1] => 599
[2,1,1,1,1,1,1] => 5295
[1,1,1,1,1,1,1,1] => 95503
[9] => 1
[8,1] => 2
[7,2] => 2
[7,1,1] => 6
[6,3] => 2
[6,2,1] => 5
[6,1,1,1] => 23
[5,4] => 2
[5,3,1] => 5
[5,2,2] => 6
[5,2,1,1] => 18
[5,1,1,1,1] => 110
[4,4,1] => 6
[4,3,2] => 5
[4,3,1,1] => 19
[4,2,2,1] => 18
[4,2,1,1,1] => 80
[4,1,1,1,1,1] => 622
[3,3,3] => 10
[3,3,2,1] => 21
[3,3,1,1,1] => 91
[3,2,2,2] => 23
[3,2,2,1,1] => 85
[3,2,1,1,1,1] => 471
[3,1,1,1,1,1,1] => 4285
[2,2,2,2,1] => 110
[2,2,2,1,1,1] => 512
[2,2,1,1,1,1,1] => 3586
[2,1,1,1,1,1,1,1] => 39468
[1,1,1,1,1,1,1,1,1] => 871030
[10] => 1
[9,1] => 2
[8,2] => 2
[8,1,1] => 6
[7,3] => 2
[7,2,1] => 5
[7,1,1,1] => 23
[6,4] => 2
[6,3,1] => 5
[6,2,2] => 6
[6,2,1,1] => 18
[6,1,1,1,1] => 110
[5,5] => 3
[5,4,1] => 6
[5,3,2] => 6
[5,3,1,1] => 18
[5,2,2,1] => 18
[5,2,1,1,1] => 78
[5,1,1,1,1,1] => 618
[4,4,2] => 6
[4,4,1,1] => 22
[4,3,3] => 6
[4,3,2,1] => 18
[4,3,1,1,1] => 84
[4,2,2,2] => 26
[4,2,2,1,1] => 78
[4,2,1,1,1,1] => 426
[4,1,1,1,1,1,1] => 4086
[3,3,3,1] => 23
[3,3,2,2] => 22
[3,3,2,1,1] => 82
[3,3,1,1,1,1] => 470
[3,2,2,2,1] => 86
[3,2,2,1,1,1] => 412
[3,2,1,1,1,1,1] => 2866
[3,1,1,1,1,1,1,1] => 32048
[2,2,2,2,2] => 246
[2,2,2,2,1,1] => 582
[2,2,2,1,1,1,1] => 3134
[2,2,1,1,1,1,1,1] => 26038
[2,1,1,1,1,1,1,1,1] => 340198
[1,1,1,1,1,1,1,1,1,1] => 8879558
[11] => 1
[10,1] => 2
[9,2] => 2
[9,1,1] => 6
[8,3] => 2
[8,2,1] => 5
[8,1,1,1] => 23
[7,4] => 2
[7,3,1] => 5
[7,2,2] => 6
[7,2,1,1] => 18
[7,1,1,1,1] => 110
[6,5] => 2
[6,4,1] => 5
[6,3,2] => 5
[6,3,1,1] => 17
[6,2,2,1] => 17
[6,2,1,1,1] => 77
[6,1,1,1,1,1] => 617
[5,5,1] => 6
[5,4,2] => 5
[5,4,1,1] => 19
[5,3,3] => 6
[5,3,2,1] => 17
[5,3,1,1,1] => 75
[5,2,2,2] => 23
[5,2,2,1,1] => 73
[5,2,1,1,1,1] => 407
[5,1,1,1,1,1,1] => 4041
[4,4,3] => 6
[4,4,2,1] => 17
[4,4,1,1,1] => 87
[4,3,3,1] => 19
[4,3,2,2] => 18
[4,3,2,1,1] => 70
[4,3,1,1,1,1] => 418
[4,2,2,2,1] => 77
[4,2,2,1,1,1] => 363
[4,2,1,1,1,1,1] => 2545
[4,1,1,1,1,1,1,1] => 30319
[3,3,3,2] => 23
[3,3,3,1,1] => 87
[3,3,2,2,1] => 79
[3,3,2,1,1,1] => 383
[3,3,1,1,1,1,1] => 2751
[3,2,2,2,2] => 110
[3,2,2,2,1,1] => 398
[3,2,2,1,1,1,1] => 2326
[3,2,1,1,1,1,1,1] => 19742
[3,1,1,1,1,1,1,1,1] => 268350
[2,2,2,2,2,1] => 617
[2,2,2,2,1,1,1] => 2855
[2,2,2,1,1,1,1,1] => 19853
[2,2,1,1,1,1,1,1,1] => 201403
[2,1,1,1,1,1,1,1,1,1] => 3166097
[1,1,1,1,1,1,1,1,1,1,1] => 98329551
[12] => 1
[11,1] => 2
[10,2] => 2
[10,1,1] => 6
[9,3] => 2
[9,2,1] => 5
[9,1,1,1] => 23
[8,4] => 2
[8,3,1] => 5
[8,2,2] => 6
[8,2,1,1] => 18
[8,1,1,1,1] => 110
[7,5] => 2
[7,4,1] => 5
[7,3,2] => 5
[7,3,1,1] => 17
[7,2,2,1] => 17
[7,2,1,1,1] => 77
[7,1,1,1,1,1] => 617
[6,6] => 3
[6,5,1] => 6
[6,4,2] => 6
[6,4,1,1] => 18
[6,3,3] => 7
[6,3,2,1] => 17
[6,3,1,1,1] => 73
[6,2,2,2] => 24
[6,2,2,1,1] => 72
[6,2,1,1,1,1] => 404
[6,1,1,1,1,1,1] => 4036
[5,5,2] => 6
[5,5,1,1] => 22
[5,4,3] => 5
[5,4,2,1] => 17
[5,4,1,1,1] => 83
[5,3,3,1] => 18
[5,3,2,2] => 19
[5,3,2,1,1] => 69
[5,3,1,1,1,1] => 387
[5,2,2,2,1] => 75
[5,2,2,1,1,1] => 349
[5,2,1,1,1,1,1] => 2451
[5,1,1,1,1,1,1,1] => 29997
[4,4,4] => 10
[4,4,3,1] => 21
[4,4,2,2] => 26
[4,4,2,1,1] => 74
[4,4,1,1,1,1] => 450
[4,3,3,2] => 18
[4,3,3,1,1] => 78
[4,3,2,2,1] => 73
[4,3,2,1,1,1] => 345
[4,3,1,1,1,1,1] => 2505
[4,2,2,2,2] => 138
[4,2,2,2,1,1] => 378
[4,2,2,1,1,1,1] => 2098
[4,2,1,1,1,1,1,1] => 17682
[4,1,1,1,1,1,1,1,1] => 253818
[3,3,3,3] => 47
[3,3,3,2,1] => 95
[3,3,3,1,1,1] => 419
[3,3,2,2,2] => 88
[3,3,2,2,1,1] => 392
[3,3,2,1,1,1,1] => 2204
[3,3,1,1,1,1,1,1] => 18908
[3,2,2,2,2,1] => 437
[3,2,2,2,1,1,1] => 2233
[3,2,2,1,1,1,1,1] => 15389
[3,2,1,1,1,1,1,1,1] => 154649
[3,1,1,1,1,1,1,1,1,1] => 2501957
[2,2,2,2,2,2] => 1602
[2,2,2,2,2,1,1] => 3546
[2,2,2,2,1,1,1,1] => 19274
[2,2,2,1,1,1,1,1,1] => 154210
[2,2,1,1,1,1,1,1,1,1] => 1807634
[2,1,1,1,1,1,1,1,1,1,1] => 33022346
[1,1,1,1,1,1,1,1,1,1,1,1] => 1191578522
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Created: May 20, 2017 at 17:57 by Martin Rubey
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Last Updated: Oct 31, 2017 at 08:07 by Martin Rubey