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Statistic identifier: St000620
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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 odd.
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 odd.
The case of an even minimum is [[St000621]].
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References:
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Code:
def statistic(L):
n = sum(L)
return sum( 1 for SYT in StandardTableaux(L) if is_odd(min( SYT.standard_descents() + [n] )) )
-----------------------------------------------------------------------------
Statistic values:
[2] => 0
[1,1] => 1
[3] => 1
[2,1] => 1
[1,1,1] => 1
[4] => 0
[3,1] => 2
[2,2] => 1
[2,1,1] => 2
[1,1,1,1] => 1
[5] => 1
[4,1] => 2
[3,2] => 3
[3,1,1] => 4
[2,2,1] => 3
[2,1,1,1] => 3
[1,1,1,1,1] => 1
[6] => 0
[5,1] => 3
[4,2] => 5
[4,1,1] => 6
[3,3] => 3
[3,2,1] => 10
[3,1,1,1] => 7
[2,2,2] => 3
[2,2,1,1] => 6
[2,1,1,1,1] => 4
[1,1,1,1,1,1] => 1
[7] => 1
[6,1] => 3
[5,2] => 8
[5,1,1] => 9
[4,3] => 8
[4,2,1] => 21
[4,1,1,1] => 13
[3,3,1] => 13
[3,2,2] => 13
[3,2,1,1] => 23
[3,1,1,1,1] => 11
[2,2,2,1] => 9
[2,2,1,1,1] => 10
[2,1,1,1,1,1] => 5
[1,1,1,1,1,1,1] => 1
[8] => 0
[7,1] => 4
[6,2] => 11
[6,1,1] => 12
[5,3] => 16
[5,2,1] => 38
[5,1,1,1] => 22
[4,4] => 8
[4,3,1] => 42
[4,2,2] => 34
[4,2,1,1] => 57
[4,1,1,1,1] => 24
[3,3,2] => 26
[3,3,1,1] => 36
[3,2,2,1] => 45
[3,2,1,1,1] => 44
[3,1,1,1,1,1] => 16
[2,2,2,2] => 9
[2,2,2,1,1] => 19
[2,2,1,1,1,1] => 15
[2,1,1,1,1,1,1] => 6
[1,1,1,1,1,1,1,1] => 1
[9] => 1
[8,1] => 4
[7,2] => 15
[7,1,1] => 16
[6,3] => 27
[6,2,1] => 61
[6,1,1,1] => 34
[5,4] => 24
[5,3,1] => 96
[5,2,2] => 72
[5,2,1,1] => 117
[5,1,1,1,1] => 46
[4,4,1] => 50
[4,3,2] => 102
[4,3,1,1] => 135
[4,2,2,1] => 136
[4,2,1,1,1] => 125
[4,1,1,1,1,1] => 40
[3,3,3] => 26
[3,3,2,1] => 107
[3,3,1,1,1] => 80
[3,2,2,2] => 54
[3,2,2,1,1] => 108
[3,2,1,1,1,1] => 75
[3,1,1,1,1,1,1] => 22
[2,2,2,2,1] => 28
[2,2,2,1,1,1] => 34
[2,2,1,1,1,1,1] => 21
[2,1,1,1,1,1,1,1] => 7
[1,1,1,1,1,1,1,1,1] => 1
[10] => 0
[9,1] => 5
[8,2] => 19
[8,1,1] => 20
[7,3] => 42
[7,2,1] => 92
[7,1,1,1] => 50
[6,4] => 51
[6,3,1] => 184
[6,2,2] => 133
[6,2,1,1] => 212
[6,1,1,1,1] => 80
[5,5] => 24
[5,4,1] => 170
[5,3,2] => 270
[5,3,1,1] => 348
[5,2,2,1] => 325
[5,2,1,1,1] => 288
[5,1,1,1,1,1] => 86
[4,4,2] => 152
[4,4,1,1] => 185
[4,3,3] => 128
[4,3,2,1] => 480
[4,3,1,1,1] => 340
[4,2,2,2] => 190
[4,2,2,1,1] => 369
[4,2,1,1,1,1] => 240
[4,1,1,1,1,1,1] => 62
[3,3,3,1] => 133
[3,3,2,2] => 161
[3,3,2,1,1] => 295
[3,3,1,1,1,1] => 155
[3,2,2,2,1] => 190
[3,2,2,1,1,1] => 217
[3,2,1,1,1,1,1] => 118
[3,1,1,1,1,1,1,1] => 29
[2,2,2,2,2] => 28
[2,2,2,2,1,1] => 62
[2,2,2,1,1,1,1] => 55
[2,2,1,1,1,1,1,1] => 28
[2,1,1,1,1,1,1,1,1] => 8
[1,1,1,1,1,1,1,1,1,1] => 1
[11] => 1
[10,1] => 5
[9,2] => 24
[9,1,1] => 25
[8,3] => 61
[8,2,1] => 131
[8,1,1,1] => 70
[7,4] => 93
[7,3,1] => 318
[7,2,2] => 225
[7,2,1,1] => 354
[7,1,1,1,1] => 130
[6,5] => 75
[6,4,1] => 405
[6,3,2] => 587
[6,3,1,1] => 744
[6,2,2,1] => 670
[6,2,1,1,1] => 580
[6,1,1,1,1,1] => 166
[5,5,1] => 194
[5,4,2] => 592
[5,4,1,1] => 703
[5,3,3] => 398
[5,3,2,1] => 1423
[5,3,1,1,1] => 976
[5,2,2,2] => 515
[5,2,2,1,1] => 982
[5,2,1,1,1,1] => 614
[5,1,1,1,1,1,1] => 148
[4,4,3] => 280
[4,4,2,1] => 817
[4,4,1,1,1] => 525
[4,3,3,1] => 741
[4,3,2,2] => 831
[4,3,2,1,1] => 1484
[4,3,1,1,1,1] => 735
[4,2,2,2,1] => 749
[4,2,2,1,1,1] => 826
[4,2,1,1,1,1,1] => 420
[4,1,1,1,1,1,1,1] => 91
[3,3,3,2] => 294
[3,3,3,1,1] => 428
[3,3,2,2,1] => 646
[3,3,2,1,1,1] => 667
[3,3,1,1,1,1,1] => 273
[3,2,2,2,2] => 218
[3,2,2,2,1,1] => 469
[3,2,2,1,1,1,1] => 390
[3,2,1,1,1,1,1,1] => 175
[3,1,1,1,1,1,1,1,1] => 37
[2,2,2,2,2,1] => 90
[2,2,2,2,1,1,1] => 117
[2,2,2,1,1,1,1,1] => 83
[2,2,1,1,1,1,1,1,1] => 36
[2,1,1,1,1,1,1,1,1,1] => 9
[1,1,1,1,1,1,1,1,1,1,1] => 1
[12] => 0
[11,1] => 6
[10,2] => 29
[10,1,1] => 30
[9,3] => 85
[9,2,1] => 180
[9,1,1,1] => 95
[8,4] => 154
[8,3,1] => 510
[8,2,2] => 356
[8,2,1,1] => 555
[8,1,1,1,1] => 200
[7,5] => 168
[7,4,1] => 816
[7,3,2] => 1130
[7,3,1,1] => 1416
[7,2,2,1] => 1249
[7,2,1,1,1] => 1064
[7,1,1,1,1,1] => 296
[6,6] => 75
[6,5,1] => 674
[6,4,2] => 1584
[6,4,1,1] => 1852
[6,3,3] => 985
[6,3,2,1] => 3424
[6,3,1,1,1] => 2300
[6,2,2,2] => 1185
[6,2,2,1,1] => 2232
[6,2,1,1,1,1] => 1360
[6,1,1,1,1,1,1] => 314
[5,5,2] => 786
[5,5,1,1] => 897
[5,4,3] => 1270
[5,4,2,1] => 3535
[5,4,1,1,1] => 2204
[5,3,3,1] => 2562
[5,3,2,2] => 2769
[5,3,2,1,1] => 4865
[5,3,1,1,1,1] => 2325
[5,2,2,2,1] => 2246
[5,2,2,1,1,1] => 2422
[5,2,1,1,1,1,1] => 1182
[5,1,1,1,1,1,1,1] => 239
[4,4,4] => 280
[4,4,3,1] => 1838
[4,4,2,2] => 1648
[4,4,2,1,1] => 2826
[4,4,1,1,1,1] => 1260
[4,3,3,2] => 1866
[4,3,3,1,1] => 2653
[4,3,2,2,1] => 3710
[4,3,2,1,1,1] => 3712
[4,3,1,1,1,1,1] => 1428
[4,2,2,2,2] => 967
[4,2,2,2,1,1] => 2044
[4,2,2,1,1,1,1] => 1636
[4,2,1,1,1,1,1,1] => 686
[4,1,1,1,1,1,1,1,1] => 128
[3,3,3,3] => 294
[3,3,3,2,1] => 1368
[3,3,3,1,1,1] => 1095
[3,3,2,2,2] => 864
[3,3,2,2,1,1] => 1782
[3,3,2,1,1,1,1] => 1330
[3,3,1,1,1,1,1,1] => 448
[3,2,2,2,2,1] => 777
[3,2,2,2,1,1,1] => 976
[3,2,2,1,1,1,1,1] => 648
[3,2,1,1,1,1,1,1,1] => 248
[3,1,1,1,1,1,1,1,1,1] => 46
[2,2,2,2,2,2] => 90
[2,2,2,2,2,1,1] => 207
[2,2,2,2,1,1,1,1] => 200
[2,2,2,1,1,1,1,1,1] => 119
[2,2,1,1,1,1,1,1,1,1] => 45
[2,1,1,1,1,1,1,1,1,1,1] => 10
[1,1,1,1,1,1,1,1,1,1,1,1] => 1
-----------------------------------------------------------------------------
Created: Oct 12, 2016 at 15:26 by Christian Stump
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Last Updated: Oct 12, 2016 at 15:34 by Christian Stump