Identifier
Identifier
Values
[2] generating graphics... => 1
[1,1] generating graphics... => 0
[3] generating graphics... => 0
[2,1] generating graphics... => 1
[1,1,1] generating graphics... => 0
[4] generating graphics... => 1
[3,1] generating graphics... => 1
[2,2] generating graphics... => 1
[2,1,1] generating graphics... => 1
[1,1,1,1] generating graphics... => 0
[5] generating graphics... => 0
[4,1] generating graphics... => 2
[3,2] generating graphics... => 2
[3,1,1] generating graphics... => 2
[2,2,1] generating graphics... => 2
[2,1,1,1] generating graphics... => 1
[1,1,1,1,1] generating graphics... => 0
[6] generating graphics... => 1
[5,1] generating graphics... => 2
[4,2] generating graphics... => 4
[4,1,1] generating graphics... => 4
[3,3] generating graphics... => 2
[3,2,1] generating graphics... => 6
[3,1,1,1] generating graphics... => 3
[2,2,2] generating graphics... => 2
[2,2,1,1] generating graphics... => 3
[2,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1] generating graphics... => 0
[7] generating graphics... => 0
[6,1] generating graphics... => 3
[5,2] generating graphics... => 6
[5,1,1] generating graphics... => 6
[4,3] generating graphics... => 6
[4,2,1] generating graphics... => 14
[4,1,1,1] generating graphics... => 7
[3,3,1] generating graphics... => 8
[3,2,2] generating graphics... => 8
[3,2,1,1] generating graphics... => 12
[3,1,1,1,1] generating graphics... => 4
[2,2,2,1] generating graphics... => 5
[2,2,1,1,1] generating graphics... => 4
[2,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1] generating graphics... => 0
[8] generating graphics... => 1
[7,1] generating graphics... => 3
[6,2] generating graphics... => 9
[6,1,1] generating graphics... => 9
[5,3] generating graphics... => 12
[5,2,1] generating graphics... => 26
[5,1,1,1] generating graphics... => 13
[4,4] generating graphics... => 6
[4,3,1] generating graphics... => 28
[4,2,2] generating graphics... => 22
[4,2,1,1] generating graphics... => 33
[4,1,1,1,1] generating graphics... => 11
[3,3,2] generating graphics... => 16
[3,3,1,1] generating graphics... => 20
[3,2,2,1] generating graphics... => 25
[3,2,1,1,1] generating graphics... => 20
[3,1,1,1,1,1] generating graphics... => 5
[2,2,2,2] generating graphics... => 5
[2,2,2,1,1] generating graphics... => 9
[2,2,1,1,1,1] generating graphics... => 5
[2,1,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1,1] generating graphics... => 0
[9] generating graphics... => 0
[8,1] generating graphics... => 4
[7,2] generating graphics... => 12
[7,1,1] generating graphics... => 12
[6,3] generating graphics... => 21
[6,2,1] generating graphics... => 44
[6,1,1,1] generating graphics... => 22
[5,4] generating graphics... => 18
[5,3,1] generating graphics... => 66
[5,2,2] generating graphics... => 48
[5,2,1,1] generating graphics... => 72
[5,1,1,1,1] generating graphics... => 24
[4,4,1] generating graphics... => 34
[4,3,2] generating graphics... => 66
[4,3,1,1] generating graphics... => 81
[4,2,2,1] generating graphics... => 80
[4,2,1,1,1] generating graphics... => 64
[4,1,1,1,1,1] generating graphics... => 16
[3,3,3] generating graphics... => 16
[3,3,2,1] generating graphics... => 61
[3,3,1,1,1] generating graphics... => 40
[3,2,2,2] generating graphics... => 30
[3,2,2,1,1] generating graphics... => 54
[3,2,1,1,1,1] generating graphics... => 30
[3,1,1,1,1,1,1] generating graphics... => 6
[2,2,2,2,1] generating graphics... => 14
[2,2,2,1,1,1] generating graphics... => 14
[2,2,1,1,1,1,1] generating graphics... => 6
[2,1,1,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1,1,1] generating graphics... => 0
[10] generating graphics... => 1
[9,1] generating graphics... => 4
[8,2] generating graphics... => 16
[8,1,1] generating graphics... => 16
[7,3] generating graphics... => 33
[7,2,1] generating graphics... => 68
[7,1,1,1] generating graphics... => 34
[6,4] generating graphics... => 39
[6,3,1] generating graphics... => 131
[6,2,2] generating graphics... => 92
[6,2,1,1] generating graphics... => 138
[6,1,1,1,1] generating graphics... => 46
[5,5] generating graphics... => 18
[5,4,1] generating graphics... => 118
[5,3,2] generating graphics... => 180
[5,3,1,1] generating graphics... => 219
[5,2,2,1] generating graphics... => 200
[5,2,1,1,1] generating graphics... => 160
[5,1,1,1,1,1] generating graphics... => 40
[4,4,2] generating graphics... => 100
[4,4,1,1] generating graphics... => 115
[4,3,3] generating graphics... => 82
[4,3,2,1] generating graphics... => 288
[4,3,1,1,1] generating graphics... => 185
[4,2,2,2] generating graphics... => 110
[4,2,2,1,1] generating graphics... => 198
[4,2,1,1,1,1] generating graphics... => 110
[4,1,1,1,1,1,1] generating graphics... => 22
[3,3,3,1] generating graphics... => 77
[3,3,2,2] generating graphics... => 91
[3,3,2,1,1] generating graphics... => 155
[3,3,1,1,1,1] generating graphics... => 70
[3,2,2,2,1] generating graphics... => 98
[3,2,2,1,1,1] generating graphics... => 98
[3,2,1,1,1,1,1] generating graphics... => 42
[3,1,1,1,1,1,1,1] generating graphics... => 7
[2,2,2,2,2] generating graphics... => 14
[2,2,2,2,1,1] generating graphics... => 28
[2,2,2,1,1,1,1] generating graphics... => 20
[2,2,1,1,1,1,1,1] generating graphics... => 7
[2,1,1,1,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1,1,1,1] generating graphics... => 0
[11] generating graphics... => 0
[10,1] generating graphics... => 5
[9,2] generating graphics... => 20
[9,1,1] generating graphics... => 20
[8,3] generating graphics... => 49
[8,2,1] generating graphics... => 100
[8,1,1,1] generating graphics... => 50
[7,4] generating graphics... => 72
[7,3,1] generating graphics... => 232
[7,2,2] generating graphics... => 160
[7,2,1,1] generating graphics... => 240
[7,1,1,1,1] generating graphics... => 80
[6,5] generating graphics... => 57
[6,4,1] generating graphics... => 288
[6,3,2] generating graphics... => 403
[6,3,1,1] generating graphics... => 488
[6,2,2,1] generating graphics... => 430
[6,2,1,1,1] generating graphics... => 344
[6,1,1,1,1,1] generating graphics... => 86
[5,5,1] generating graphics... => 136
[5,4,2] generating graphics... => 398
[5,4,1,1] generating graphics... => 452
[5,3,3] generating graphics... => 262
[5,3,2,1] generating graphics... => 887
[5,3,1,1,1] generating graphics... => 564
[5,2,2,2] generating graphics... => 310
[5,2,2,1,1] generating graphics... => 558
[5,2,1,1,1,1] generating graphics... => 310
[5,1,1,1,1,1,1] generating graphics... => 62
[4,4,3] generating graphics... => 182
[4,4,2,1] generating graphics... => 503
[4,4,1,1,1] generating graphics... => 300
[4,3,3,1] generating graphics... => 447
[4,3,2,2] generating graphics... => 489
[4,3,2,1,1] generating graphics... => 826
[4,3,1,1,1,1] generating graphics... => 365
[4,2,2,2,1] generating graphics... => 406
[4,2,2,1,1,1] generating graphics... => 406
[4,2,1,1,1,1,1] generating graphics... => 174
[4,1,1,1,1,1,1,1] generating graphics... => 29
[3,3,3,2] generating graphics... => 168
[3,3,3,1,1] generating graphics... => 232
[3,3,2,2,1] generating graphics... => 344
[3,3,2,1,1,1] generating graphics... => 323
[3,3,1,1,1,1,1] generating graphics... => 112
[3,2,2,2,2] generating graphics... => 112
[3,2,2,2,1,1] generating graphics... => 224
[3,2,2,1,1,1,1] generating graphics... => 160
[3,2,1,1,1,1,1,1] generating graphics... => 56
[3,1,1,1,1,1,1,1,1] generating graphics... => 8
[2,2,2,2,2,1] generating graphics... => 42
[2,2,2,2,1,1,1] generating graphics... => 48
[2,2,2,1,1,1,1,1] generating graphics... => 27
[2,2,1,1,1,1,1,1,1] generating graphics... => 8
[2,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 0
[12] generating graphics... => 1
[11,1] generating graphics... => 5
[10,2] generating graphics... => 25
[10,1,1] generating graphics... => 25
[9,3] generating graphics... => 69
[9,2,1] generating graphics... => 140
[9,1,1,1] generating graphics... => 70
[8,4] generating graphics... => 121
[8,3,1] generating graphics... => 381
[8,2,2] generating graphics... => 260
[8,2,1,1] generating graphics... => 390
[8,1,1,1,1] generating graphics... => 130
[7,5] generating graphics... => 129
[7,4,1] generating graphics... => 592
[7,3,2] generating graphics... => 795
[7,3,1,1] generating graphics... => 960
[7,2,2,1] generating graphics... => 830
[7,2,1,1,1] generating graphics... => 664
[7,1,1,1,1,1] generating graphics... => 166
[6,6] generating graphics... => 57
[6,5,1] generating graphics... => 481
[6,4,2] generating graphics... => 1089
[6,4,1,1] generating graphics... => 1228
[6,3,3] generating graphics... => 665
[6,3,2,1] generating graphics... => 2208
[6,3,1,1,1] generating graphics... => 1396
[6,2,2,2] generating graphics... => 740
[6,2,2,1,1] generating graphics... => 1332
[6,2,1,1,1,1] generating graphics... => 740
[6,1,1,1,1,1,1] generating graphics... => 148
[5,5,2] generating graphics... => 534
[5,5,1,1] generating graphics... => 588
[5,4,3] generating graphics... => 842
[5,4,2,1] generating graphics... => 2240
[5,4,1,1,1] generating graphics... => 1316
[5,3,3,1] generating graphics... => 1596
[5,3,2,2] generating graphics... => 1686
[5,3,2,1,1] generating graphics... => 2835
[5,3,1,1,1,1] generating graphics... => 1239
[5,2,2,2,1] generating graphics... => 1274
[5,2,2,1,1,1] generating graphics... => 1274
[5,2,1,1,1,1,1] generating graphics... => 546
[5,1,1,1,1,1,1,1] generating graphics... => 91
[4,4,4] generating graphics... => 182
[4,4,3,1] generating graphics... => 1132
[4,4,2,2] generating graphics... => 992
[4,4,2,1,1] generating graphics... => 1629
[4,4,1,1,1,1] generating graphics... => 665
[4,3,3,2] generating graphics... => 1104
[4,3,3,1,1] generating graphics... => 1505
[4,3,2,2,1] generating graphics... => 2065
[4,3,2,1,1,1] generating graphics... => 1920
[4,3,1,1,1,1,1] generating graphics... => 651
[4,2,2,2,2] generating graphics... => 518
[4,2,2,2,1,1] generating graphics... => 1036
[4,2,2,1,1,1,1] generating graphics... => 740
[4,2,1,1,1,1,1,1] generating graphics... => 259
[4,1,1,1,1,1,1,1,1] generating graphics... => 37
[3,3,3,3] generating graphics... => 168
[3,3,3,2,1] generating graphics... => 744
[3,3,3,1,1,1] generating graphics... => 555
[3,3,2,2,2] generating graphics... => 456
[3,3,2,2,1,1] generating graphics... => 891
[3,3,2,1,1,1,1] generating graphics... => 595
[3,3,1,1,1,1,1,1] generating graphics... => 168
[3,2,2,2,2,1] generating graphics... => 378
[3,2,2,2,1,1,1] generating graphics... => 432
[3,2,2,1,1,1,1,1] generating graphics... => 243
[3,2,1,1,1,1,1,1,1] generating graphics... => 72
[3,1,1,1,1,1,1,1,1,1] generating graphics... => 9
[2,2,2,2,2,2] generating graphics... => 42
[2,2,2,2,2,1,1] generating graphics... => 90
[2,2,2,2,1,1,1,1] generating graphics... => 75
[2,2,2,1,1,1,1,1,1] generating graphics... => 35
[2,2,1,1,1,1,1,1,1,1] generating graphics... => 9
[2,1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 0
click to show generating function       
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 St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd..
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
Code
def statistic(L):
    n = sum(L)
    return sum(1 for SYT in StandardTableaux(L) if is_even(min( SYT.standard_descents() + [n] )) )

Created
Oct 12, 2016 at 15:26 by Christian Stump
Updated
Oct 12, 2016 at 15:35 by Christian Stump