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Statistic identifier: St000621
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Collection: Integer partitions
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Description: The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even.
This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1].
The case of an odd minimum is [[St000620]].
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References: [1] Reiner, V., Webb, P. The combinatorics of the bar resolution in group cohomology [[MathSciNet:2043333]]
[2] Athanasiadis, C. A. The symmetric group action on rank-selected posets of injective words [[arXiv:1606.03829]]
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Code:
def statistic(L):
n = sum(L)
return sum(1 for SYT in StandardTableaux(L) if is_even(min( SYT.standard_descents() + [n] )) )
-----------------------------------------------------------------------------
Statistic values:
[2] => 1
[1,1] => 0
[3] => 0
[2,1] => 1
[1,1,1] => 0
[4] => 1
[3,1] => 1
[2,2] => 1
[2,1,1] => 1
[1,1,1,1] => 0
[5] => 0
[4,1] => 2
[3,2] => 2
[3,1,1] => 2
[2,2,1] => 2
[2,1,1,1] => 1
[1,1,1,1,1] => 0
[6] => 1
[5,1] => 2
[4,2] => 4
[4,1,1] => 4
[3,3] => 2
[3,2,1] => 6
[3,1,1,1] => 3
[2,2,2] => 2
[2,2,1,1] => 3
[2,1,1,1,1] => 1
[1,1,1,1,1,1] => 0
[7] => 0
[6,1] => 3
[5,2] => 6
[5,1,1] => 6
[4,3] => 6
[4,2,1] => 14
[4,1,1,1] => 7
[3,3,1] => 8
[3,2,2] => 8
[3,2,1,1] => 12
[3,1,1,1,1] => 4
[2,2,2,1] => 5
[2,2,1,1,1] => 4
[2,1,1,1,1,1] => 1
[1,1,1,1,1,1,1] => 0
[8] => 1
[7,1] => 3
[6,2] => 9
[6,1,1] => 9
[5,3] => 12
[5,2,1] => 26
[5,1,1,1] => 13
[4,4] => 6
[4,3,1] => 28
[4,2,2] => 22
[4,2,1,1] => 33
[4,1,1,1,1] => 11
[3,3,2] => 16
[3,3,1,1] => 20
[3,2,2,1] => 25
[3,2,1,1,1] => 20
[3,1,1,1,1,1] => 5
[2,2,2,2] => 5
[2,2,2,1,1] => 9
[2,2,1,1,1,1] => 5
[2,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1] => 0
[9] => 0
[8,1] => 4
[7,2] => 12
[7,1,1] => 12
[6,3] => 21
[6,2,1] => 44
[6,1,1,1] => 22
[5,4] => 18
[5,3,1] => 66
[5,2,2] => 48
[5,2,1,1] => 72
[5,1,1,1,1] => 24
[4,4,1] => 34
[4,3,2] => 66
[4,3,1,1] => 81
[4,2,2,1] => 80
[4,2,1,1,1] => 64
[4,1,1,1,1,1] => 16
[3,3,3] => 16
[3,3,2,1] => 61
[3,3,1,1,1] => 40
[3,2,2,2] => 30
[3,2,2,1,1] => 54
[3,2,1,1,1,1] => 30
[3,1,1,1,1,1,1] => 6
[2,2,2,2,1] => 14
[2,2,2,1,1,1] => 14
[2,2,1,1,1,1,1] => 6
[2,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1] => 0
[10] => 1
[9,1] => 4
[8,2] => 16
[8,1,1] => 16
[7,3] => 33
[7,2,1] => 68
[7,1,1,1] => 34
[6,4] => 39
[6,3,1] => 131
[6,2,2] => 92
[6,2,1,1] => 138
[6,1,1,1,1] => 46
[5,5] => 18
[5,4,1] => 118
[5,3,2] => 180
[5,3,1,1] => 219
[5,2,2,1] => 200
[5,2,1,1,1] => 160
[5,1,1,1,1,1] => 40
[4,4,2] => 100
[4,4,1,1] => 115
[4,3,3] => 82
[4,3,2,1] => 288
[4,3,1,1,1] => 185
[4,2,2,2] => 110
[4,2,2,1,1] => 198
[4,2,1,1,1,1] => 110
[4,1,1,1,1,1,1] => 22
[3,3,3,1] => 77
[3,3,2,2] => 91
[3,3,2,1,1] => 155
[3,3,1,1,1,1] => 70
[3,2,2,2,1] => 98
[3,2,2,1,1,1] => 98
[3,2,1,1,1,1,1] => 42
[3,1,1,1,1,1,1,1] => 7
[2,2,2,2,2] => 14
[2,2,2,2,1,1] => 28
[2,2,2,1,1,1,1] => 20
[2,2,1,1,1,1,1,1] => 7
[2,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1] => 0
[11] => 0
[10,1] => 5
[9,2] => 20
[9,1,1] => 20
[8,3] => 49
[8,2,1] => 100
[8,1,1,1] => 50
[7,4] => 72
[7,3,1] => 232
[7,2,2] => 160
[7,2,1,1] => 240
[7,1,1,1,1] => 80
[6,5] => 57
[6,4,1] => 288
[6,3,2] => 403
[6,3,1,1] => 488
[6,2,2,1] => 430
[6,2,1,1,1] => 344
[6,1,1,1,1,1] => 86
[5,5,1] => 136
[5,4,2] => 398
[5,4,1,1] => 452
[5,3,3] => 262
[5,3,2,1] => 887
[5,3,1,1,1] => 564
[5,2,2,2] => 310
[5,2,2,1,1] => 558
[5,2,1,1,1,1] => 310
[5,1,1,1,1,1,1] => 62
[4,4,3] => 182
[4,4,2,1] => 503
[4,4,1,1,1] => 300
[4,3,3,1] => 447
[4,3,2,2] => 489
[4,3,2,1,1] => 826
[4,3,1,1,1,1] => 365
[4,2,2,2,1] => 406
[4,2,2,1,1,1] => 406
[4,2,1,1,1,1,1] => 174
[4,1,1,1,1,1,1,1] => 29
[3,3,3,2] => 168
[3,3,3,1,1] => 232
[3,3,2,2,1] => 344
[3,3,2,1,1,1] => 323
[3,3,1,1,1,1,1] => 112
[3,2,2,2,2] => 112
[3,2,2,2,1,1] => 224
[3,2,2,1,1,1,1] => 160
[3,2,1,1,1,1,1,1] => 56
[3,1,1,1,1,1,1,1,1] => 8
[2,2,2,2,2,1] => 42
[2,2,2,2,1,1,1] => 48
[2,2,2,1,1,1,1,1] => 27
[2,2,1,1,1,1,1,1,1] => 8
[2,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1] => 0
[12] => 1
[11,1] => 5
[10,2] => 25
[10,1,1] => 25
[9,3] => 69
[9,2,1] => 140
[9,1,1,1] => 70
[8,4] => 121
[8,3,1] => 381
[8,2,2] => 260
[8,2,1,1] => 390
[8,1,1,1,1] => 130
[7,5] => 129
[7,4,1] => 592
[7,3,2] => 795
[7,3,1,1] => 960
[7,2,2,1] => 830
[7,2,1,1,1] => 664
[7,1,1,1,1,1] => 166
[6,6] => 57
[6,5,1] => 481
[6,4,2] => 1089
[6,4,1,1] => 1228
[6,3,3] => 665
[6,3,2,1] => 2208
[6,3,1,1,1] => 1396
[6,2,2,2] => 740
[6,2,2,1,1] => 1332
[6,2,1,1,1,1] => 740
[6,1,1,1,1,1,1] => 148
[5,5,2] => 534
[5,5,1,1] => 588
[5,4,3] => 842
[5,4,2,1] => 2240
[5,4,1,1,1] => 1316
[5,3,3,1] => 1596
[5,3,2,2] => 1686
[5,3,2,1,1] => 2835
[5,3,1,1,1,1] => 1239
[5,2,2,2,1] => 1274
[5,2,2,1,1,1] => 1274
[5,2,1,1,1,1,1] => 546
[5,1,1,1,1,1,1,1] => 91
[4,4,4] => 182
[4,4,3,1] => 1132
[4,4,2,2] => 992
[4,4,2,1,1] => 1629
[4,4,1,1,1,1] => 665
[4,3,3,2] => 1104
[4,3,3,1,1] => 1505
[4,3,2,2,1] => 2065
[4,3,2,1,1,1] => 1920
[4,3,1,1,1,1,1] => 651
[4,2,2,2,2] => 518
[4,2,2,2,1,1] => 1036
[4,2,2,1,1,1,1] => 740
[4,2,1,1,1,1,1,1] => 259
[4,1,1,1,1,1,1,1,1] => 37
[3,3,3,3] => 168
[3,3,3,2,1] => 744
[3,3,3,1,1,1] => 555
[3,3,2,2,2] => 456
[3,3,2,2,1,1] => 891
[3,3,2,1,1,1,1] => 595
[3,3,1,1,1,1,1,1] => 168
[3,2,2,2,2,1] => 378
[3,2,2,2,1,1,1] => 432
[3,2,2,1,1,1,1,1] => 243
[3,2,1,1,1,1,1,1,1] => 72
[3,1,1,1,1,1,1,1,1,1] => 9
[2,2,2,2,2,2] => 42
[2,2,2,2,2,1,1] => 90
[2,2,2,2,1,1,1,1] => 75
[2,2,2,1,1,1,1,1,1] => 35
[2,2,1,1,1,1,1,1,1,1] => 9
[2,1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1,1] => 0
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Created: Oct 12, 2016 at 15:26 by Christian Stump
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Last Updated: Oct 12, 2016 at 15:35 by Christian Stump