Identifier
Values
[2] => [1,1] => 0
[1,1] => [2] => 1
[3] => [1,1,1] => 0
[2,1] => [2,1] => 1
[1,1,1] => [3] => 0
[4] => [1,1,1,1] => 0
[3,1] => [2,1,1] => 1
[2,2] => [2,2] => 1
[2,1,1] => [3,1] => 1
[1,1,1,1] => [4] => 1
[5] => [1,1,1,1,1] => 0
[4,1] => [2,1,1,1] => 1
[3,2] => [2,2,1] => 2
[3,1,1] => [3,1,1] => 2
[2,2,1] => [3,2] => 2
[2,1,1,1] => [4,1] => 2
[1,1,1,1,1] => [5] => 0
[6] => [1,1,1,1,1,1] => 0
[5,1] => [2,1,1,1,1] => 1
[4,2] => [2,2,1,1] => 3
[4,1,1] => [3,1,1,1] => 3
[3,3] => [2,2,2] => 2
[3,2,1] => [3,2,1] => 6
[3,1,1,1] => [4,1,1] => 4
[2,2,2] => [3,3] => 2
[2,2,1,1] => [4,2] => 4
[2,1,1,1,1] => [5,1] => 2
[1,1,1,1,1,1] => [6] => 1
[7] => [1,1,1,1,1,1,1] => 0
[6,1] => [2,1,1,1,1,1] => 1
[5,2] => [2,2,1,1,1] => 4
[5,1,1] => [3,1,1,1,1] => 4
[4,3] => [2,2,2,1] => 5
[4,2,1] => [3,2,1,1] => 12
[4,1,1,1] => [4,1,1,1] => 7
[3,3,1] => [3,2,2] => 8
[3,2,2] => [3,3,1] => 8
[3,2,1,1] => [4,2,1] => 14
[3,1,1,1,1] => [5,1,1] => 6
[2,2,2,1] => [4,3] => 6
[2,2,1,1,1] => [5,2] => 6
[2,1,1,1,1,1] => [6,1] => 3
[1,1,1,1,1,1,1] => [7] => 0
[8] => [1,1,1,1,1,1,1,1] => 0
[7,1] => [2,1,1,1,1,1,1] => 1
[6,2] => [2,2,1,1,1,1] => 5
[6,1,1] => [3,1,1,1,1,1] => 5
[5,3] => [2,2,2,1,1] => 9
[5,2,1] => [3,2,1,1,1] => 20
[5,1,1,1] => [4,1,1,1,1] => 11
[4,4] => [2,2,2,2] => 5
[4,3,1] => [3,2,2,1] => 25
[4,2,2] => [3,3,1,1] => 20
[4,2,1,1] => [4,2,1,1] => 33
[4,1,1,1,1] => [5,1,1,1] => 13
[3,3,2] => [3,3,2] => 16
[3,3,1,1] => [4,2,2] => 22
[3,2,2,1] => [4,3,1] => 28
[3,2,1,1,1] => [5,2,1] => 26
[3,1,1,1,1,1] => [6,1,1] => 9
[2,2,2,2] => [4,4] => 6
[2,2,2,1,1] => [5,3] => 12
[2,2,1,1,1,1] => [6,2] => 9
[2,1,1,1,1,1,1] => [7,1] => 3
[1,1,1,1,1,1,1,1] => [8] => 1
[9] => [1,1,1,1,1,1,1,1,1] => 0
[8,1] => [2,1,1,1,1,1,1,1] => 1
[7,2] => [2,2,1,1,1,1,1] => 6
[7,1,1] => [3,1,1,1,1,1,1] => 6
[6,3] => [2,2,2,1,1,1] => 14
[6,2,1] => [3,2,1,1,1,1] => 30
[6,1,1,1] => [4,1,1,1,1,1] => 16
[5,4] => [2,2,2,2,1] => 14
[5,3,1] => [3,2,2,1,1] => 54
[5,2,2] => [3,3,1,1,1] => 40
[5,2,1,1] => [4,2,1,1,1] => 64
[5,1,1,1,1] => [5,1,1,1,1] => 24
[4,4,1] => [3,2,2,2] => 30
[4,3,2] => [3,3,2,1] => 61
[4,3,1,1] => [4,2,2,1] => 80
[4,2,2,1] => [4,3,1,1] => 81
[4,2,1,1,1] => [5,2,1,1] => 72
[4,1,1,1,1,1] => [6,1,1,1] => 22
[3,3,3] => [3,3,3] => 16
[3,3,2,1] => [4,3,2] => 66
[3,3,1,1,1] => [5,2,2] => 48
[3,2,2,2] => [4,4,1] => 34
[3,2,2,1,1] => [5,3,1] => 66
[3,2,1,1,1,1] => [6,2,1] => 44
[3,1,1,1,1,1,1] => [7,1,1] => 12
[2,2,2,2,1] => [5,4] => 18
[2,2,2,1,1,1] => [6,3] => 21
[2,2,1,1,1,1,1] => [7,2] => 12
[2,1,1,1,1,1,1,1] => [8,1] => 4
[1,1,1,1,1,1,1,1,1] => [9] => 0
[10] => [1,1,1,1,1,1,1,1,1,1] => 0
[9,1] => [2,1,1,1,1,1,1,1,1] => 1
[8,2] => [2,2,1,1,1,1,1,1] => 7
[8,1,1] => [3,1,1,1,1,1,1,1] => 7
[7,3] => [2,2,2,1,1,1,1] => 20
[7,2,1] => [3,2,1,1,1,1,1] => 42
>>> Load all 270 entries. <<<
[7,1,1,1] => [4,1,1,1,1,1,1] => 22
[6,4] => [2,2,2,2,1,1] => 28
[6,3,1] => [3,2,2,1,1,1] => 98
[6,2,2] => [3,3,1,1,1,1] => 70
[6,2,1,1] => [4,2,1,1,1,1] => 110
[6,1,1,1,1] => [5,1,1,1,1,1] => 40
[5,5] => [2,2,2,2,2] => 14
[5,4,1] => [3,2,2,2,1] => 98
[5,3,2] => [3,3,2,1,1] => 155
[5,3,1,1] => [4,2,2,1,1] => 198
[5,2,2,1] => [4,3,1,1,1] => 185
[5,2,1,1,1] => [5,2,1,1,1] => 160
[5,1,1,1,1,1] => [6,1,1,1,1] => 46
[4,4,2] => [3,3,2,2] => 91
[4,4,1,1] => [4,2,2,2] => 110
[4,3,3] => [3,3,3,1] => 77
[4,3,2,1] => [4,3,2,1] => 288
[4,3,1,1,1] => [5,2,2,1] => 200
[4,2,2,2] => [4,4,1,1] => 115
[4,2,2,1,1] => [5,3,1,1] => 219
[4,2,1,1,1,1] => [6,2,1,1] => 138
[4,1,1,1,1,1,1] => [7,1,1,1] => 34
[3,3,3,1] => [4,3,3] => 82
[3,3,2,2] => [4,4,2] => 100
[3,3,2,1,1] => [5,3,2] => 180
[3,3,1,1,1,1] => [6,2,2] => 92
[3,2,2,2,1] => [5,4,1] => 118
[3,2,2,1,1,1] => [6,3,1] => 131
[3,2,1,1,1,1,1] => [7,2,1] => 68
[3,1,1,1,1,1,1,1] => [8,1,1] => 16
[2,2,2,2,2] => [5,5] => 18
[2,2,2,2,1,1] => [6,4] => 39
[2,2,2,1,1,1,1] => [7,3] => 33
[2,2,1,1,1,1,1,1] => [8,2] => 16
[2,1,1,1,1,1,1,1,1] => [9,1] => 4
[1,1,1,1,1,1,1,1,1,1] => [10] => 1
[11] => [1,1,1,1,1,1,1,1,1,1,1] => 0
[10,1] => [2,1,1,1,1,1,1,1,1,1] => 1
[9,2] => [2,2,1,1,1,1,1,1,1] => 8
[9,1,1] => [3,1,1,1,1,1,1,1,1] => 8
[8,3] => [2,2,2,1,1,1,1,1] => 27
[8,2,1] => [3,2,1,1,1,1,1,1] => 56
[8,1,1,1] => [4,1,1,1,1,1,1,1] => 29
[7,4] => [2,2,2,2,1,1,1] => 48
[7,3,1] => [3,2,2,1,1,1,1] => 160
[7,2,2] => [3,3,1,1,1,1,1] => 112
[7,2,1,1] => [4,2,1,1,1,1,1] => 174
[7,1,1,1,1] => [5,1,1,1,1,1,1] => 62
[6,5] => [2,2,2,2,2,1] => 42
[6,4,1] => [3,2,2,2,1,1] => 224
[6,3,2] => [3,3,2,1,1,1] => 323
[6,3,1,1] => [4,2,2,1,1,1] => 406
[6,2,2,1] => [4,3,1,1,1,1] => 365
[6,2,1,1,1] => [5,2,1,1,1,1] => 310
[6,1,1,1,1,1] => [6,1,1,1,1,1] => 86
[5,5,1] => [3,2,2,2,2] => 112
[5,4,2] => [3,3,2,2,1] => 344
[5,4,1,1] => [4,2,2,2,1] => 406
[5,3,3] => [3,3,3,1,1] => 232
[5,3,2,1] => [4,3,2,1,1] => 826
[5,3,1,1,1] => [5,2,2,1,1] => 558
[5,2,2,2] => [4,4,1,1,1] => 300
[5,2,2,1,1] => [5,3,1,1,1] => 564
[5,2,1,1,1,1] => [6,2,1,1,1] => 344
[5,1,1,1,1,1,1] => [7,1,1,1,1] => 80
[4,4,3] => [3,3,3,2] => 168
[4,4,2,1] => [4,3,2,2] => 489
[4,4,1,1,1] => [5,2,2,2] => 310
[4,3,3,1] => [4,3,3,1] => 447
[4,3,2,2] => [4,4,2,1] => 503
[4,3,2,1,1] => [5,3,2,1] => 887
[4,3,1,1,1,1] => [6,2,2,1] => 430
[4,2,2,2,1] => [5,4,1,1] => 452
[4,2,2,1,1,1] => [6,3,1,1] => 488
[4,2,1,1,1,1,1] => [7,2,1,1] => 240
[4,1,1,1,1,1,1,1] => [8,1,1,1] => 50
[3,3,3,2] => [4,4,3] => 182
[3,3,3,1,1] => [5,3,3] => 262
[3,3,2,2,1] => [5,4,2] => 398
[3,3,2,1,1,1] => [6,3,2] => 403
[3,3,1,1,1,1,1] => [7,2,2] => 160
[3,2,2,2,2] => [5,5,1] => 136
[3,2,2,2,1,1] => [6,4,1] => 288
[3,2,2,1,1,1,1] => [7,3,1] => 232
[3,2,1,1,1,1,1,1] => [8,2,1] => 100
[3,1,1,1,1,1,1,1,1] => [9,1,1] => 20
[2,2,2,2,2,1] => [6,5] => 57
[2,2,2,2,1,1,1] => [7,4] => 72
[2,2,2,1,1,1,1,1] => [8,3] => 49
[2,2,1,1,1,1,1,1,1] => [9,2] => 20
[2,1,1,1,1,1,1,1,1,1] => [10,1] => 5
[1,1,1,1,1,1,1,1,1,1,1] => [11] => 0
[12] => [1,1,1,1,1,1,1,1,1,1,1,1] => 0
[11,1] => [2,1,1,1,1,1,1,1,1,1,1] => 1
[10,2] => [2,2,1,1,1,1,1,1,1,1] => 9
[10,1,1] => [3,1,1,1,1,1,1,1,1,1] => 9
[9,3] => [2,2,2,1,1,1,1,1,1] => 35
[9,2,1] => [3,2,1,1,1,1,1,1,1] => 72
[9,1,1,1] => [4,1,1,1,1,1,1,1,1] => 37
[8,4] => [2,2,2,2,1,1,1,1] => 75
[8,3,1] => [3,2,2,1,1,1,1,1] => 243
[8,2,2] => [3,3,1,1,1,1,1,1] => 168
[8,2,1,1] => [4,2,1,1,1,1,1,1] => 259
[8,1,1,1,1] => [5,1,1,1,1,1,1,1] => 91
[7,5] => [2,2,2,2,2,1,1] => 90
[7,4,1] => [3,2,2,2,1,1,1] => 432
[7,3,2] => [3,3,2,1,1,1,1] => 595
[7,3,1,1] => [4,2,2,1,1,1,1] => 740
[7,2,2,1] => [4,3,1,1,1,1,1] => 651
[7,2,1,1,1] => [5,2,1,1,1,1,1] => 546
[7,1,1,1,1,1] => [6,1,1,1,1,1,1] => 148
[6,6] => [2,2,2,2,2,2] => 42
[6,5,1] => [3,2,2,2,2,1] => 378
[6,4,2] => [3,3,2,2,1,1] => 891
[6,4,1,1] => [4,2,2,2,1,1] => 1036
[6,3,3] => [3,3,3,1,1,1] => 555
[6,3,2,1] => [4,3,2,1,1,1] => 1920
[6,3,1,1,1] => [5,2,2,1,1,1] => 1274
[6,2,2,2] => [4,4,1,1,1,1] => 665
[6,2,2,1,1] => [5,3,1,1,1,1] => 1239
[6,2,1,1,1,1] => [6,2,1,1,1,1] => 740
[6,1,1,1,1,1,1] => [7,1,1,1,1,1] => 166
[5,5,2] => [3,3,2,2,2] => 456
[5,5,1,1] => [4,2,2,2,2] => 518
[5,4,3] => [3,3,3,2,1] => 744
[5,4,2,1] => [4,3,2,2,1] => 2065
[5,4,1,1,1] => [5,2,2,2,1] => 1274
[5,3,3,1] => [4,3,3,1,1] => 1505
[5,3,2,2] => [4,4,2,1,1] => 1629
[5,3,2,1,1] => [5,3,2,1,1] => 2835
[5,3,1,1,1,1] => [6,2,2,1,1] => 1332
[5,2,2,2,1] => [5,4,1,1,1] => 1316
[5,2,2,1,1,1] => [6,3,1,1,1] => 1396
[5,2,1,1,1,1,1] => [7,2,1,1,1] => 664
[5,1,1,1,1,1,1,1] => [8,1,1,1,1] => 130
[4,4,4] => [3,3,3,3] => 168
[4,4,3,1] => [4,3,3,2] => 1104
[4,4,2,2] => [4,4,2,2] => 992
[4,4,2,1,1] => [5,3,2,2] => 1686
[4,4,1,1,1,1] => [6,2,2,2] => 740
[4,3,3,2] => [4,4,3,1] => 1132
[4,3,3,1,1] => [5,3,3,1] => 1596
[4,3,2,2,1] => [5,4,2,1] => 2240
[4,3,2,1,1,1] => [6,3,2,1] => 2208
[4,3,1,1,1,1,1] => [7,2,2,1] => 830
[4,2,2,2,2] => [5,5,1,1] => 588
[4,2,2,2,1,1] => [6,4,1,1] => 1228
[4,2,2,1,1,1,1] => [7,3,1,1] => 960
[4,2,1,1,1,1,1,1] => [8,2,1,1] => 390
[4,1,1,1,1,1,1,1,1] => [9,1,1,1] => 70
[3,3,3,3] => [4,4,4] => 182
[3,3,3,2,1] => [5,4,3] => 842
[3,3,3,1,1,1] => [6,3,3] => 665
[3,3,2,2,2] => [5,5,2] => 534
[3,3,2,2,1,1] => [6,4,2] => 1089
[3,3,2,1,1,1,1] => [7,3,2] => 795
[3,3,1,1,1,1,1,1] => [8,2,2] => 260
[3,2,2,2,2,1] => [6,5,1] => 481
[3,2,2,2,1,1,1] => [7,4,1] => 592
[3,2,2,1,1,1,1,1] => [8,3,1] => 381
[3,2,1,1,1,1,1,1,1] => [9,2,1] => 140
[3,1,1,1,1,1,1,1,1,1] => [10,1,1] => 25
[2,2,2,2,2,2] => [6,6] => 57
[2,2,2,2,2,1,1] => [7,5] => 129
[2,2,2,2,1,1,1,1] => [8,4] => 121
[2,2,2,1,1,1,1,1,1] => [9,3] => 69
[2,2,1,1,1,1,1,1,1,1] => [10,2] => 25
[2,1,1,1,1,1,1,1,1,1,1] => [11,1] => 5
[1,1,1,1,1,1,1,1,1,1,1,1] => [12] => 1
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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 St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd..
Map
conjugate
Description
Return the conjugate partition of the partition.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.