Identifier
Values
[1] => [1] => 0
[2] => [1,1] => 1
[1,1] => [2] => 0
[3] => [1,1,1] => 0
[2,1] => [2,1] => 1
[1,1,1] => [3] => 0
[4] => [1,1,1,1] => 1
[3,1] => [2,1,1] => 1
[2,2] => [2,2] => 1
[2,1,1] => [3,1] => 1
[1,1,1,1] => [4] => 0
[5] => [1,1,1,1,1] => 0
[4,1] => [2,1,1,1] => 2
[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] => 1
[1,1,1,1,1] => [5] => 0
[6] => [1,1,1,1,1,1] => 1
[5,1] => [2,1,1,1,1] => 2
[4,2] => [2,2,1,1] => 4
[4,1,1] => [3,1,1,1] => 4
[3,3] => [2,2,2] => 2
[3,2,1] => [3,2,1] => 6
[3,1,1,1] => [4,1,1] => 3
[2,2,2] => [3,3] => 2
[2,2,1,1] => [4,2] => 3
[2,1,1,1,1] => [5,1] => 1
[1,1,1,1,1,1] => [6] => 0
[7] => [1,1,1,1,1,1,1] => 0
[6,1] => [2,1,1,1,1,1] => 3
[5,2] => [2,2,1,1,1] => 6
[5,1,1] => [3,1,1,1,1] => 6
[4,3] => [2,2,2,1] => 6
[4,2,1] => [3,2,1,1] => 14
[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] => 12
[3,1,1,1,1] => [5,1,1] => 4
[2,2,2,1] => [4,3] => 5
[2,2,1,1,1] => [5,2] => 4
[2,1,1,1,1,1] => [6,1] => 1
[1,1,1,1,1,1,1] => [7] => 0
[8] => [1,1,1,1,1,1,1,1] => 1
[7,1] => [2,1,1,1,1,1,1] => 3
[6,2] => [2,2,1,1,1,1] => 9
[6,1,1] => [3,1,1,1,1,1] => 9
[5,3] => [2,2,2,1,1] => 12
[5,2,1] => [3,2,1,1,1] => 26
[5,1,1,1] => [4,1,1,1,1] => 13
[4,4] => [2,2,2,2] => 6
[4,3,1] => [3,2,2,1] => 28
[4,2,2] => [3,3,1,1] => 22
[4,2,1,1] => [4,2,1,1] => 33
[4,1,1,1,1] => [5,1,1,1] => 11
[3,3,2] => [3,3,2] => 16
[3,3,1,1] => [4,2,2] => 20
[3,2,2,1] => [4,3,1] => 25
[3,2,1,1,1] => [5,2,1] => 20
[3,1,1,1,1,1] => [6,1,1] => 5
[2,2,2,2] => [4,4] => 5
[2,2,2,1,1] => [5,3] => 9
[2,2,1,1,1,1] => [6,2] => 5
[2,1,1,1,1,1,1] => [7,1] => 1
[1,1,1,1,1,1,1,1] => [8] => 0
[9] => [1,1,1,1,1,1,1,1,1] => 0
[8,1] => [2,1,1,1,1,1,1,1] => 4
[7,2] => [2,2,1,1,1,1,1] => 12
[7,1,1] => [3,1,1,1,1,1,1] => 12
[6,3] => [2,2,2,1,1,1] => 21
[6,2,1] => [3,2,1,1,1,1] => 44
[6,1,1,1] => [4,1,1,1,1,1] => 22
[5,4] => [2,2,2,2,1] => 18
[5,3,1] => [3,2,2,1,1] => 66
[5,2,2] => [3,3,1,1,1] => 48
[5,2,1,1] => [4,2,1,1,1] => 72
[5,1,1,1,1] => [5,1,1,1,1] => 24
[4,4,1] => [3,2,2,2] => 34
[4,3,2] => [3,3,2,1] => 66
[4,3,1,1] => [4,2,2,1] => 81
[4,2,2,1] => [4,3,1,1] => 80
[4,2,1,1,1] => [5,2,1,1] => 64
[4,1,1,1,1,1] => [6,1,1,1] => 16
[3,3,3] => [3,3,3] => 16
[3,3,2,1] => [4,3,2] => 61
[3,3,1,1,1] => [5,2,2] => 40
[3,2,2,2] => [4,4,1] => 30
[3,2,2,1,1] => [5,3,1] => 54
[3,2,1,1,1,1] => [6,2,1] => 30
[3,1,1,1,1,1,1] => [7,1,1] => 6
[2,2,2,2,1] => [5,4] => 14
[2,2,2,1,1,1] => [6,3] => 14
[2,2,1,1,1,1,1] => [7,2] => 6
[2,1,1,1,1,1,1,1] => [8,1] => 1
[1,1,1,1,1,1,1,1,1] => [9] => 0
[10] => [1,1,1,1,1,1,1,1,1,1] => 1
[9,1] => [2,1,1,1,1,1,1,1,1] => 4
[8,2] => [2,2,1,1,1,1,1,1] => 16
[8,1,1] => [3,1,1,1,1,1,1,1] => 16
[7,3] => [2,2,2,1,1,1,1] => 33
>>> Load all 271 entries. <<<
[7,2,1] => [3,2,1,1,1,1,1] => 68
[7,1,1,1] => [4,1,1,1,1,1,1] => 34
[6,4] => [2,2,2,2,1,1] => 39
[6,3,1] => [3,2,2,1,1,1] => 131
[6,2,2] => [3,3,1,1,1,1] => 92
[6,2,1,1] => [4,2,1,1,1,1] => 138
[6,1,1,1,1] => [5,1,1,1,1,1] => 46
[5,5] => [2,2,2,2,2] => 18
[5,4,1] => [3,2,2,2,1] => 118
[5,3,2] => [3,3,2,1,1] => 180
[5,3,1,1] => [4,2,2,1,1] => 219
[5,2,2,1] => [4,3,1,1,1] => 200
[5,2,1,1,1] => [5,2,1,1,1] => 160
[5,1,1,1,1,1] => [6,1,1,1,1] => 40
[4,4,2] => [3,3,2,2] => 100
[4,4,1,1] => [4,2,2,2] => 115
[4,3,3] => [3,3,3,1] => 82
[4,3,2,1] => [4,3,2,1] => 288
[4,3,1,1,1] => [5,2,2,1] => 185
[4,2,2,2] => [4,4,1,1] => 110
[4,2,2,1,1] => [5,3,1,1] => 198
[4,2,1,1,1,1] => [6,2,1,1] => 110
[4,1,1,1,1,1,1] => [7,1,1,1] => 22
[3,3,3,1] => [4,3,3] => 77
[3,3,2,2] => [4,4,2] => 91
[3,3,2,1,1] => [5,3,2] => 155
[3,3,1,1,1,1] => [6,2,2] => 70
[3,2,2,2,1] => [5,4,1] => 98
[3,2,2,1,1,1] => [6,3,1] => 98
[3,2,1,1,1,1,1] => [7,2,1] => 42
[3,1,1,1,1,1,1,1] => [8,1,1] => 7
[2,2,2,2,2] => [5,5] => 14
[2,2,2,2,1,1] => [6,4] => 28
[2,2,2,1,1,1,1] => [7,3] => 20
[2,2,1,1,1,1,1,1] => [8,2] => 7
[2,1,1,1,1,1,1,1,1] => [9,1] => 1
[1,1,1,1,1,1,1,1,1,1] => [10] => 0
[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] => 5
[9,2] => [2,2,1,1,1,1,1,1,1] => 20
[9,1,1] => [3,1,1,1,1,1,1,1,1] => 20
[8,3] => [2,2,2,1,1,1,1,1] => 49
[8,2,1] => [3,2,1,1,1,1,1,1] => 100
[8,1,1,1] => [4,1,1,1,1,1,1,1] => 50
[7,4] => [2,2,2,2,1,1,1] => 72
[7,3,1] => [3,2,2,1,1,1,1] => 232
[7,2,2] => [3,3,1,1,1,1,1] => 160
[7,2,1,1] => [4,2,1,1,1,1,1] => 240
[7,1,1,1,1] => [5,1,1,1,1,1,1] => 80
[6,5] => [2,2,2,2,2,1] => 57
[6,4,1] => [3,2,2,2,1,1] => 288
[6,3,2] => [3,3,2,1,1,1] => 403
[6,3,1,1] => [4,2,2,1,1,1] => 488
[6,2,2,1] => [4,3,1,1,1,1] => 430
[6,2,1,1,1] => [5,2,1,1,1,1] => 344
[6,1,1,1,1,1] => [6,1,1,1,1,1] => 86
[5,5,1] => [3,2,2,2,2] => 136
[5,4,2] => [3,3,2,2,1] => 398
[5,4,1,1] => [4,2,2,2,1] => 452
[5,3,3] => [3,3,3,1,1] => 262
[5,3,2,1] => [4,3,2,1,1] => 887
[5,3,1,1,1] => [5,2,2,1,1] => 564
[5,2,2,2] => [4,4,1,1,1] => 310
[5,2,2,1,1] => [5,3,1,1,1] => 558
[5,2,1,1,1,1] => [6,2,1,1,1] => 310
[5,1,1,1,1,1,1] => [7,1,1,1,1] => 62
[4,4,3] => [3,3,3,2] => 182
[4,4,2,1] => [4,3,2,2] => 503
[4,4,1,1,1] => [5,2,2,2] => 300
[4,3,3,1] => [4,3,3,1] => 447
[4,3,2,2] => [4,4,2,1] => 489
[4,3,2,1,1] => [5,3,2,1] => 826
[4,3,1,1,1,1] => [6,2,2,1] => 365
[4,2,2,2,1] => [5,4,1,1] => 406
[4,2,2,1,1,1] => [6,3,1,1] => 406
[4,2,1,1,1,1,1] => [7,2,1,1] => 174
[4,1,1,1,1,1,1,1] => [8,1,1,1] => 29
[3,3,3,2] => [4,4,3] => 168
[3,3,3,1,1] => [5,3,3] => 232
[3,3,2,2,1] => [5,4,2] => 344
[3,3,2,1,1,1] => [6,3,2] => 323
[3,3,1,1,1,1,1] => [7,2,2] => 112
[3,2,2,2,2] => [5,5,1] => 112
[3,2,2,2,1,1] => [6,4,1] => 224
[3,2,2,1,1,1,1] => [7,3,1] => 160
[3,2,1,1,1,1,1,1] => [8,2,1] => 56
[3,1,1,1,1,1,1,1,1] => [9,1,1] => 8
[2,2,2,2,2,1] => [6,5] => 42
[2,2,2,2,1,1,1] => [7,4] => 48
[2,2,2,1,1,1,1,1] => [8,3] => 27
[2,2,1,1,1,1,1,1,1] => [9,2] => 8
[2,1,1,1,1,1,1,1,1,1] => [10,1] => 1
[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] => 1
[11,1] => [2,1,1,1,1,1,1,1,1,1,1] => 5
[10,2] => [2,2,1,1,1,1,1,1,1,1] => 25
[10,1,1] => [3,1,1,1,1,1,1,1,1,1] => 25
[9,3] => [2,2,2,1,1,1,1,1,1] => 69
[9,2,1] => [3,2,1,1,1,1,1,1,1] => 140
[9,1,1,1] => [4,1,1,1,1,1,1,1,1] => 70
[8,4] => [2,2,2,2,1,1,1,1] => 121
[8,3,1] => [3,2,2,1,1,1,1,1] => 381
[8,2,2] => [3,3,1,1,1,1,1,1] => 260
[8,2,1,1] => [4,2,1,1,1,1,1,1] => 390
[8,1,1,1,1] => [5,1,1,1,1,1,1,1] => 130
[7,5] => [2,2,2,2,2,1,1] => 129
[7,4,1] => [3,2,2,2,1,1,1] => 592
[7,3,2] => [3,3,2,1,1,1,1] => 795
[7,3,1,1] => [4,2,2,1,1,1,1] => 960
[7,2,2,1] => [4,3,1,1,1,1,1] => 830
[7,2,1,1,1] => [5,2,1,1,1,1,1] => 664
[7,1,1,1,1,1] => [6,1,1,1,1,1,1] => 166
[6,6] => [2,2,2,2,2,2] => 57
[6,5,1] => [3,2,2,2,2,1] => 481
[6,4,2] => [3,3,2,2,1,1] => 1089
[6,4,1,1] => [4,2,2,2,1,1] => 1228
[6,3,3] => [3,3,3,1,1,1] => 665
[6,3,2,1] => [4,3,2,1,1,1] => 2208
[6,3,1,1,1] => [5,2,2,1,1,1] => 1396
[6,2,2,2] => [4,4,1,1,1,1] => 740
[6,2,2,1,1] => [5,3,1,1,1,1] => 1332
[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] => 148
[5,5,2] => [3,3,2,2,2] => 534
[5,5,1,1] => [4,2,2,2,2] => 588
[5,4,3] => [3,3,3,2,1] => 842
[5,4,2,1] => [4,3,2,2,1] => 2240
[5,4,1,1,1] => [5,2,2,2,1] => 1316
[5,3,3,1] => [4,3,3,1,1] => 1596
[5,3,2,2] => [4,4,2,1,1] => 1686
[5,3,2,1,1] => [5,3,2,1,1] => 2835
[5,3,1,1,1,1] => [6,2,2,1,1] => 1239
[5,2,2,2,1] => [5,4,1,1,1] => 1274
[5,2,2,1,1,1] => [6,3,1,1,1] => 1274
[5,2,1,1,1,1,1] => [7,2,1,1,1] => 546
[5,1,1,1,1,1,1,1] => [8,1,1,1,1] => 91
[4,4,4] => [3,3,3,3] => 182
[4,4,3,1] => [4,3,3,2] => 1132
[4,4,2,2] => [4,4,2,2] => 992
[4,4,2,1,1] => [5,3,2,2] => 1629
[4,4,1,1,1,1] => [6,2,2,2] => 665
[4,3,3,2] => [4,4,3,1] => 1104
[4,3,3,1,1] => [5,3,3,1] => 1505
[4,3,2,2,1] => [5,4,2,1] => 2065
[4,3,2,1,1,1] => [6,3,2,1] => 1920
[4,3,1,1,1,1,1] => [7,2,2,1] => 651
[4,2,2,2,2] => [5,5,1,1] => 518
[4,2,2,2,1,1] => [6,4,1,1] => 1036
[4,2,2,1,1,1,1] => [7,3,1,1] => 740
[4,2,1,1,1,1,1,1] => [8,2,1,1] => 259
[4,1,1,1,1,1,1,1,1] => [9,1,1,1] => 37
[3,3,3,3] => [4,4,4] => 168
[3,3,3,2,1] => [5,4,3] => 744
[3,3,3,1,1,1] => [6,3,3] => 555
[3,3,2,2,2] => [5,5,2] => 456
[3,3,2,2,1,1] => [6,4,2] => 891
[3,3,2,1,1,1,1] => [7,3,2] => 595
[3,3,1,1,1,1,1,1] => [8,2,2] => 168
[3,2,2,2,2,1] => [6,5,1] => 378
[3,2,2,2,1,1,1] => [7,4,1] => 432
[3,2,2,1,1,1,1,1] => [8,3,1] => 243
[3,2,1,1,1,1,1,1,1] => [9,2,1] => 72
[3,1,1,1,1,1,1,1,1,1] => [10,1,1] => 9
[2,2,2,2,2,2] => [6,6] => 42
[2,2,2,2,2,1,1] => [7,5] => 90
[2,2,2,2,1,1,1,1] => [8,4] => 75
[2,2,2,1,1,1,1,1,1] => [9,3] => 35
[2,2,1,1,1,1,1,1,1,1] => [10,2] => 9
[2,1,1,1,1,1,1,1,1,1,1] => [11,1] => 1
[1,1,1,1,1,1,1,1,1,1,1,1] => [12] => 0
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Description
The number of standard desarrangement tableaux of shape equal to the given partition.
A standard desarrangement tableau is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry $i$ such that $i+1$ appears to the right or above $i$ in the tableau (with respect to English tableau notation).
This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also:
  • St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition.: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition
  • St000500Eigenvalues of the random-to-random operator acting on the regular representation.: Eigenvalues of the random-to-random operator acting on the regular representation.
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.