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-----------------------------------------------------------------------------
Statistic identifier: St000814
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions.
For example, $e_{22} = s_{1111} + s_{211} + s_{22}$, so the statistic on the partition $22$ is $3$.
-----------------------------------------------------------------------------
References: [1] a(n)= sum of entries of n-th Kostka matrix for the partitions of n. [[OEIS:A178718]]
-----------------------------------------------------------------------------
Code:
def statistic(mu):
s = SymmetricFunctions(ZZ).s()
e = SymmetricFunctions(ZZ).e()
return sum(coeff for _, coeff in s(e(mu)))
-----------------------------------------------------------------------------
Statistic values:
[] => 1
[1] => 1
[2] => 1
[1,1] => 2
[3] => 1
[2,1] => 2
[1,1,1] => 4
[4] => 1
[3,1] => 2
[2,2] => 3
[2,1,1] => 5
[1,1,1,1] => 10
[5] => 1
[4,1] => 2
[3,2] => 3
[3,1,1] => 5
[2,2,1] => 7
[2,1,1,1] => 13
[1,1,1,1,1] => 26
[6] => 1
[5,1] => 2
[4,2] => 3
[4,1,1] => 5
[3,3] => 4
[3,2,1] => 8
[3,1,1,1] => 14
[2,2,2] => 11
[2,2,1,1] => 20
[2,1,1,1,1] => 38
[1,1,1,1,1,1] => 76
[7] => 1
[6,1] => 2
[5,2] => 3
[5,1,1] => 5
[4,3] => 4
[4,2,1] => 8
[4,1,1,1] => 14
[3,3,1] => 10
[3,2,2] => 13
[3,2,1,1] => 23
[3,1,1,1,1] => 42
[2,2,2,1] => 32
[2,2,1,1,1] => 60
[2,1,1,1,1,1] => 116
[1,1,1,1,1,1,1] => 232
[8] => 1
[7,1] => 2
[6,2] => 3
[6,1,1] => 5
[5,3] => 4
[5,2,1] => 8
[5,1,1,1] => 14
[4,4] => 5
[4,3,1] => 11
[4,2,2] => 14
[4,2,1,1] => 24
[4,1,1,1,1] => 43
[3,3,2] => 17
[3,3,1,1] => 30
[3,2,2,1] => 40
[3,2,1,1,1] => 73
[3,1,1,1,1,1] => 136
[2,2,2,2] => 56
[2,2,2,1,1] => 103
[2,2,1,1,1,1] => 196
[2,1,1,1,1,1,1] => 382
[1,1,1,1,1,1,1,1] => 764
[9] => 1
[8,1] => 2
[7,2] => 3
[7,1,1] => 5
[6,3] => 4
[6,2,1] => 8
[6,1,1,1] => 14
[5,4] => 5
[5,3,1] => 11
[5,2,2] => 14
[5,2,1,1] => 24
[5,1,1,1,1] => 43
[4,4,1] => 13
[4,3,2] => 19
[4,3,1,1] => 33
[4,2,2,1] => 43
[4,2,1,1,1] => 77
[4,1,1,1,1,1] => 141
[3,3,3] => 23
[3,3,2,1] => 53
[3,3,1,1,1] => 96
[3,2,2,2] => 72
[3,2,2,1,1] => 131
[3,2,1,1,1,1] => 244
[3,1,1,1,1,1,1] => 462
[2,2,2,2,1] => 184
[2,2,2,1,1,1] => 347
[2,2,1,1,1,1,1] => 668
[2,1,1,1,1,1,1,1] => 1310
[1,1,1,1,1,1,1,1,1] => 2620
[10] => 1
[9,1] => 2
[8,2] => 3
[8,1,1] => 5
[7,3] => 4
[7,2,1] => 8
[7,1,1,1] => 14
[6,4] => 5
[6,3,1] => 11
[6,2,2] => 14
[6,2,1,1] => 24
[6,1,1,1,1] => 43
[5,5] => 6
[5,4,1] => 14
[5,3,2] => 20
[5,3,1,1] => 34
[5,2,2,1] => 44
[5,2,1,1,1] => 78
[5,1,1,1,1,1] => 142
[4,4,2] => 23
[4,4,1,1] => 40
[4,3,3] => 27
[4,3,2,1] => 61
[4,3,1,1,1] => 109
[4,2,2,2] => 81
[4,2,2,1,1] => 145
[4,2,1,1,1,1] => 265
[4,1,1,1,1,1,1] => 492
[3,3,3,1] => 74
[3,3,2,2] => 100
[3,3,2,1,1] => 180
[3,3,1,1,1,1] => 332
[3,2,2,2,1] => 248
[3,2,2,1,1,1] => 460
[3,2,1,1,1,1,1] => 868
[3,1,1,1,1,1,1,1] => 1660
[2,2,2,2,2] => 348
[2,2,2,2,1,1] => 652
[2,2,2,1,1,1,1] => 1244
[2,2,1,1,1,1,1,1] => 2412
[2,1,1,1,1,1,1,1,1] => 4748
[1,1,1,1,1,1,1,1,1,1] => 9496
[11] => 1
[10,1] => 2
[9,2] => 3
[9,1,1] => 5
[8,3] => 4
[8,2,1] => 8
[8,1,1,1] => 14
[7,4] => 5
[7,3,1] => 11
[7,2,2] => 14
[7,2,1,1] => 24
[7,1,1,1,1] => 43
[6,5] => 6
[6,4,1] => 14
[6,3,2] => 20
[6,3,1,1] => 34
[6,2,2,1] => 44
[6,2,1,1,1] => 78
[6,1,1,1,1,1] => 142
[5,5,1] => 16
[5,4,2] => 25
[5,4,1,1] => 43
[5,3,3] => 29
[5,3,2,1] => 64
[5,3,1,1,1] => 113
[5,2,2,2] => 84
[5,2,2,1,1] => 149
[5,2,1,1,1,1] => 270
[5,1,1,1,1,1,1] => 498
[4,4,3] => 33
[4,4,2,1] => 74
[4,4,1,1,1] => 132
[4,3,3,1] => 88
[4,3,2,2] => 117
[4,3,2,1,1] => 209
[4,3,1,1,1,1] => 381
[4,2,2,2,1] => 282
[4,2,2,1,1,1] => 516
[4,2,1,1,1,1,1] => 958
[4,1,1,1,1,1,1,1] => 1800
[3,3,3,2] => 143
[3,3,3,1,1] => 256
[3,3,2,2,1] => 350
[3,3,2,1,1,1] => 644
[3,3,1,1,1,1,1] => 1204
[3,2,2,2,2] => 484
[3,2,2,2,1,1] => 897
[3,2,2,1,1,1,1] => 1688
[3,2,1,1,1,1,1,1] => 3218
[3,1,1,1,1,1,1,1,1] => 6204
[2,2,2,2,2,1] => 1268
[2,2,2,2,1,1,1] => 2408
[2,2,2,1,1,1,1,1] => 4636
[2,2,1,1,1,1,1,1,1] => 9040
[2,1,1,1,1,1,1,1,1,1] => 17848
[1,1,1,1,1,1,1,1,1,1,1] => 35696
[12] => 1
[11,1] => 2
[10,2] => 3
[10,1,1] => 5
[9,3] => 4
[9,2,1] => 8
[9,1,1,1] => 14
[8,4] => 5
[8,3,1] => 11
[8,2,2] => 14
[8,2,1,1] => 24
[8,1,1,1,1] => 43
[7,5] => 6
[7,4,1] => 14
[7,3,2] => 20
[7,3,1,1] => 34
[7,2,2,1] => 44
[7,2,1,1,1] => 78
[7,1,1,1,1,1] => 142
[6,6] => 7
[6,5,1] => 17
[6,4,2] => 26
[6,4,1,1] => 44
[6,3,3] => 30
[6,3,2,1] => 65
[6,3,1,1,1] => 114
[6,2,2,2] => 85
[6,2,2,1,1] => 150
[6,2,1,1,1,1] => 271
[6,1,1,1,1,1,1] => 499
[5,5,2] => 29
[5,5,1,1] => 50
[5,4,3] => 37
[5,4,2,1] => 82
[5,4,1,1,1] => 145
[5,3,3,1] => 96
[5,3,2,2] => 126
[5,3,2,1,1] => 223
[5,3,1,1,1,1] => 402
[5,2,2,2,1] => 297
[5,2,2,1,1,1] => 538
[5,2,1,1,1,1,1] => 989
[5,1,1,1,1,1,1,1] => 1842
[4,4,4] => 42
[4,4,3,1] => 110
[4,4,2,2] => 146
[4,4,2,1,1] => 259
[4,4,1,1,1,1] => 470
[4,3,3,2] => 175
[4,3,3,1,1] => 311
[4,3,2,2,1] => 420
[4,3,2,1,1,1] => 765
[4,3,1,1,1,1,1] => 1414
[4,2,2,2,2] => 572
[4,2,2,2,1,1] => 1046
[4,2,2,1,1,1,1] => 1941
[4,2,1,1,1,1,1,1] => 3643
[4,1,1,1,1,1,1,1,1] => 6904
[3,3,3,3] => 214
[3,3,3,2,1] => 517
[3,3,3,1,1,1] => 944
[3,3,2,2,2] => 710
[3,3,2,2,1,1] => 1306
[3,3,2,1,1,1,1] => 2436
[3,3,1,1,1,1,1,1] => 4600
[3,2,2,2,2,1] => 1824
[3,2,2,2,1,1,1] => 3425
[3,2,2,1,1,1,1,1] => 6508
[3,2,1,1,1,1,1,1,1] => 12498
[3,1,1,1,1,1,1,1,1,1] => 24232
[2,2,2,2,2,2] => 2578
[2,2,2,2,2,1,1] => 4876
[2,2,2,2,1,1,1,1] => 9340
[2,2,2,1,1,1,1,1,1] => 18092
[2,2,1,1,1,1,1,1,1,1] => 35420
[2,1,1,1,1,1,1,1,1,1,1] => 70076
[1,1,1,1,1,1,1,1,1,1,1,1] => 140152
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Created: May 20, 2017 at 17:44 by Martin Rubey
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Last Updated: Oct 31, 2017 at 08:10 by Martin Rubey