Identifier
Values
[1] => 1
[2] => 1
[1,1] => 1
[3] => 1
[2,1] => 1
[1,1,1] => 1
[4] => 1
[3,1] => 1
[2,2] => 1
[2,1,1] => 1
[1,1,1,1] => 1
[5] => 1
[4,1] => 2
[3,2] => 1
[3,1,1] => 1
[2,2,1] => 1
[2,1,1,1] => 1
[1,1,1,1,1] => 1
[6] => 3
[5,1] => 3
[4,2] => 2
[4,1,1] => 2
[3,3] => 1
[3,2,1] => 3
[3,1,1,1] => 1
[2,2,2] => 1
[2,2,1,1] => 1
[2,1,1,1,1] => 1
[1,1,1,1,1,1] => 1
[7] => 5
[6,1] => 6
[5,2] => 4
[5,1,1] => 4
[4,3] => 3
[4,2,1] => 5
[4,1,1,1] => 2
[3,3,1] => 2
[3,2,2] => 2
[3,2,1,1] => 3
[3,1,1,1,1] => 1
[2,2,2,1] => 1
[2,2,1,1,1] => 1
[2,1,1,1,1,1] => 1
[1,1,1,1,1,1,1] => 1
[8] => 12
[7,1] => 13
[6,2] => 7
[6,1,1] => 8
[5,3] => 6
[5,2,1] => 9
[5,1,1,1] => 4
[4,4] => 3
[4,3,1] => 8
[4,2,2] => 3
[4,2,1,1] => 6
[4,1,1,1,1] => 2
[3,3,2] => 3
[3,3,1,1] => 3
[3,2,2,1] => 4
[3,2,1,1,1] => 3
[3,1,1,1,1,1] => 1
[2,2,2,2] => 1
[2,2,2,1,1] => 1
[2,2,1,1,1,1] => 1
[2,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1] => 1
[9] => 24
[8,1] => 27
[7,2] => 16
[7,1,1] => 16
[6,3] => 13
[6,2,1] => 18
[6,1,1,1] => 8
[5,4] => 11
[5,3,1] => 15
[5,2,2] => 6
[5,2,1,1] => 11
[5,1,1,1,1] => 4
[4,4,1] => 7
[4,3,2] => 10
[4,3,1,1] => 10
[4,2,2,1] => 8
[4,2,1,1,1] => 6
[4,1,1,1,1,1] => 2
[3,3,3] => 2
[3,3,2,1] => 7
[3,3,1,1,1] => 3
[3,2,2,2] => 2
[3,2,2,1,1] => 5
[3,2,1,1,1,1] => 3
[3,1,1,1,1,1,1] => 1
[2,2,2,2,1] => 1
[2,2,2,1,1,1] => 1
[2,2,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1] => 1
[10] => 77
[9,1] => 86
[8,2] => 48
[8,1,1] => 48
[7,3] => 39
>>> Load all 913 entries. <<<[7,2,1] => 55
[7,1,1,1] => 18
[6,4] => 32
[6,3,1] => 45
[6,2,2] => 16
[6,2,1,1] => 32
[6,1,1,1,1] => 8
[5,5] => 15
[5,4,1] => 38
[5,3,2] => 27
[5,3,1,1] => 27
[5,2,2,1] => 19
[5,2,1,1,1] => 12
[5,1,1,1,1,1] => 4
[4,4,2] => 12
[4,4,1,1] => 12
[4,3,3] => 12
[4,3,2,1] => 34
[4,3,1,1,1] => 11
[4,2,2,2] => 4
[4,2,2,1,1] => 12
[4,2,1,1,1,1] => 6
[4,1,1,1,1,1,1] => 2
[3,3,3,1] => 5
[3,3,2,2] => 6
[3,3,2,1,1] => 12
[3,3,1,1,1,1] => 3
[3,2,2,2,1] => 6
[3,2,2,1,1,1] => 5
[3,2,1,1,1,1,1] => 3
[3,1,1,1,1,1,1,1] => 1
[2,2,2,2,2] => 1
[2,2,2,2,1,1] => 1
[2,2,2,1,1,1,1] => 1
[2,2,1,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1] => 1
[11] => 210
[10,1] => 231
[9,2] => 138
[9,1,1] => 138
[8,3] => 99
[8,2,1] => 155
[8,1,1,1] => 56
[7,4] => 83
[7,3,1] => 113
[7,2,2] => 46
[7,2,1,1] => 83
[7,1,1,1,1] => 18
[6,5] => 77
[6,4,1] => 97
[6,3,2] => 71
[6,3,1,1] => 71
[6,2,2,1] => 54
[6,2,1,1,1] => 36
[6,1,1,1,1,1] => 8
[5,5,1] => 46
[5,4,2] => 62
[5,4,1,1] => 62
[5,3,3] => 27
[5,3,2,1] => 85
[5,3,1,1,1] => 31
[5,2,2,2] => 11
[5,2,2,1,1] => 30
[5,2,1,1,1,1] => 12
[5,1,1,1,1,1,1] => 4
[4,4,3] => 25
[4,4,2,1] => 39
[4,4,1,1,1] => 14
[4,3,3,1] => 34
[4,3,2,2] => 28
[4,3,2,1,1] => 50
[4,3,1,1,1,1] => 11
[4,2,2,2,1] => 14
[4,2,2,1,1,1] => 13
[4,2,1,1,1,1,1] => 6
[4,1,1,1,1,1,1,1] => 2
[3,3,3,2] => 8
[3,3,3,1,1] => 8
[3,3,2,2,1] => 19
[3,3,2,1,1,1] => 13
[3,3,1,1,1,1,1] => 3
[3,2,2,2,2] => 2
[3,2,2,2,1,1] => 9
[3,2,2,1,1,1,1] => 5
[3,2,1,1,1,1,1,1] => 3
[3,1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2,1] => 1
[2,2,2,2,1,1,1] => 1
[2,2,2,1,1,1,1,1] => 1
[2,2,1,1,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1] => 1
[12] => 640
[11,1] => 700
[10,2] => 386
[10,1,1] => 384
[9,3] => 288
[9,2,1] => 428
[9,1,1,1] => 150
[8,4] => 246
[8,3,1] => 324
[8,2,2] => 113
[8,2,1,1] => 246
[8,1,1,1,1] => 56
[7,5] => 220
[7,4,1] => 280
[7,3,2] => 185
[7,3,1,1] => 185
[7,2,2,1] => 130
[7,2,1,1,1] => 95
[7,1,1,1,1,1] => 18
[6,6] => 109
[6,5,1] => 257
[6,4,2] => 163
[6,4,1,1] => 164
[6,3,3] => 72
[6,3,2,1] => 216
[6,3,1,1,1] => 76
[6,2,2,2] => 27
[6,2,2,1,1] => 78
[6,2,1,1,1,1] => 36
[6,1,1,1,1,1,1] => 8
[5,5,2] => 76
[5,5,1,1] => 78
[5,4,3] => 132
[5,4,2,1] => 196
[5,4,1,1,1] => 67
[5,3,3,1] => 87
[5,3,2,2] => 61
[5,3,2,1,1] => 132
[5,3,1,1,1,1] => 31
[5,2,2,2,1] => 32
[5,2,2,1,1,1] => 33
[5,2,1,1,1,1,1] => 12
[5,1,1,1,1,1,1,1] => 4
[4,4,4] => 20
[4,4,3,1] => 84
[4,4,2,2] => 31
[4,4,2,1,1] => 62
[4,4,1,1,1,1] => 14
[4,3,3,2] => 56
[4,3,3,1,1] => 55
[4,3,2,2,1] => 78
[4,3,2,1,1,1] => 57
[4,3,1,1,1,1,1] => 11
[4,2,2,2,2] => 5
[4,2,2,2,1,1] => 19
[4,2,2,1,1,1,1] => 13
[4,2,1,1,1,1,1,1] => 6
[4,1,1,1,1,1,1,1,1] => 2
[3,3,3,3] => 4
[3,3,3,2,1] => 24
[3,3,3,1,1,1] => 8
[3,3,2,2,2] => 9
[3,3,2,2,1,1] => 27
[3,3,2,1,1,1,1] => 13
[3,3,1,1,1,1,1,1] => 3
[3,2,2,2,2,1] => 7
[3,2,2,2,1,1,1] => 9
[3,2,2,1,1,1,1,1] => 5
[3,2,1,1,1,1,1,1,1] => 3
[3,1,1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2,2] => 1
[2,2,2,2,2,1,1] => 1
[2,2,2,2,1,1,1,1] => 1
[2,2,2,1,1,1,1,1,1] => 1
[2,2,1,1,1,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1,1] => 1
[13] => 1650
[12,1] => 1790
[11,2] => 1000
[11,1,1] => 1000
[10,3] => 730
[10,2,1] => 1098
[10,1,1,1] => 420
[9,4] => 601
[9,3,1] => 811
[9,2,2] => 308
[9,2,1,1] => 601
[9,1,1,1,1] => 150
[8,5] => 536
[8,4,1] => 675
[8,3,2] => 466
[8,3,1,1] => 466
[8,2,2,1] => 347
[8,2,1,1,1] => 263
[8,1,1,1,1,1] => 56
[7,6] => 513
[7,5,1] => 613
[7,4,2] => 399
[7,4,1,1] => 399
[7,3,3] => 180
[7,3,2,1] => 533
[7,3,1,1,1] => 200
[7,2,2,2] => 74
[7,2,2,1,1] => 194
[7,2,1,1,1,1] => 95
[7,1,1,1,1,1,1] => 18
[6,6,1] => 297
[6,5,2] => 365
[6,5,1,1] => 368
[6,4,3] => 308
[6,4,2,1] => 465
[6,4,1,1,1] => 179
[6,3,3,1] => 210
[6,3,2,2] => 156
[6,3,2,1,1] => 308
[6,3,1,1,1,1] => 76
[6,2,2,2,1] => 87
[6,2,2,1,1,1] => 82
[6,2,1,1,1,1,1] => 36
[6,1,1,1,1,1,1,1] => 8
[5,5,3] => 146
[5,5,2,1] => 222
[5,5,1,1,1] => 84
[5,4,4] => 137
[5,4,3,1] => 368
[5,4,2,2] => 142
[5,4,2,1,1] => 273
[5,4,1,1,1,1] => 67
[5,3,3,2] => 128
[5,3,3,1,1] => 125
[5,3,2,2,1] => 187
[5,3,2,1,1,1] => 140
[5,3,1,1,1,1,1] => 31
[5,2,2,2,2] => 14
[5,2,2,2,1,1] => 46
[5,2,2,1,1,1,1] => 33
[5,2,1,1,1,1,1,1] => 12
[5,1,1,1,1,1,1,1,1] => 4
[4,4,4,1] => 59
[4,4,3,2] => 120
[4,4,3,1,1] => 120
[4,4,2,2,1] => 90
[4,4,2,1,1,1] => 67
[4,4,1,1,1,1,1] => 14
[4,3,3,3] => 37
[4,3,3,2,1] => 161
[4,3,3,1,1,1] => 59
[4,3,2,2,2] => 43
[4,3,2,2,1,1] => 116
[4,3,2,1,1,1,1] => 57
[4,3,1,1,1,1,1,1] => 11
[4,2,2,2,2,1] => 17
[4,2,2,2,1,1,1] => 19
[4,2,2,1,1,1,1,1] => 13
[4,2,1,1,1,1,1,1,1] => 6
[4,1,1,1,1,1,1,1,1,1] => 2
[3,3,3,3,1] => 12
[3,3,3,2,2] => 18
[3,3,3,2,1,1] => 37
[3,3,3,1,1,1,1] => 8
[3,3,2,2,2,1] => 29
[3,3,2,2,1,1,1] => 27
[3,3,2,1,1,1,1,1] => 13
[3,3,1,1,1,1,1,1,1] => 3
[3,2,2,2,2,2] => 3
[3,2,2,2,2,1,1] => 10
[3,2,2,2,1,1,1,1] => 9
[3,2,2,1,1,1,1,1,1] => 5
[3,2,1,1,1,1,1,1,1,1] => 3
[3,1,1,1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2,2,1] => 1
[2,2,2,2,2,1,1,1] => 1
[2,2,2,2,1,1,1,1,1] => 1
[2,2,2,1,1,1,1,1,1,1] => 1
[2,2,1,1,1,1,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1,1,1] => 1
[14] => 4973
[13,1] => 5346
[12,2] => 3126
[12,1,1] => 3126
[11,3] => 2140
[11,2,1] => 3402
[11,1,1,1] => 1296
[10,4] => 1753
[10,3,1] => 2354
[10,2,2] => 966
[10,2,1,1] => 1753
[10,1,1,1,1] => 420
[9,5] => 1548
[9,4,1] => 1944
[9,3,2] => 1386
[9,3,1,1] => 1386
[9,2,2,1] => 1080
[9,2,1,1,1] => 756
[9,1,1,1,1,1] => 150
[8,6] => 1452
[8,5,1] => 1743
[8,4,2] => 1179
[8,4,1,1] => 1179
[8,3,3] => 504
[8,3,2,1] => 1560
[8,3,1,1,1] => 594
[8,2,2,2] => 204
[8,2,2,1,1] => 576
[8,2,1,1,1,1] => 263
[8,1,1,1,1,1,1] => 56
[7,7] => 718
[7,6,1] => 1661
[7,5,2] => 1071
[7,5,1,1] => 1071
[7,4,3] => 844
[7,4,2,1] => 1350
[7,4,1,1,1] => 517
[7,3,3,1] => 577
[7,3,2,2] => 463
[7,3,2,1,1] => 844
[7,3,1,1,1,1] => 200
[7,2,2,2,1] => 229
[7,2,2,1,1,1] => 221
[7,2,1,1,1,1,1] => 95
[7,1,1,1,1,1,1,1] => 18
[6,6,2] => 507
[6,6,1,1] => 515
[6,5,3] => 783
[6,5,2,1] => 1244
[6,5,1,1,1] => 478
[6,4,4] => 371
[6,4,3,1] => 983
[6,4,2,2] => 412
[6,4,2,1,1] => 738
[6,4,1,1,1,1] => 179
[6,3,3,2] => 352
[6,3,3,1,1] => 346
[6,3,2,2,1] => 538
[6,3,2,1,1,1] => 380
[6,3,1,1,1,1,1] => 76
[6,2,2,2,2] => 37
[6,2,2,2,1,1] => 132
[6,2,2,1,1,1,1] => 82
[6,2,1,1,1,1,1,1] => 36
[6,1,1,1,1,1,1,1,1] => 8
[5,5,4] => 355
[5,5,3,1] => 470
[5,5,2,2] => 201
[5,5,2,1,1] => 355
[5,5,1,1,1,1] => 84
[5,4,4,1] => 440
[5,4,3,2] => 631
[5,4,3,1,1] => 631
[5,4,2,2,1] => 494
[5,4,2,1,1,1] => 344
[5,4,1,1,1,1,1] => 67
[5,3,3,3] => 92
[5,3,3,2,1] => 422
[5,3,3,1,1,1] => 156
[5,3,2,2,2] => 107
[5,3,2,2,1,1] => 309
[5,3,2,1,1,1,1] => 140
[5,3,1,1,1,1,1,1] => 31
[5,2,2,2,2,1] => 44
[5,2,2,2,1,1,1] => 50
[5,2,2,1,1,1,1,1] => 33
[5,2,1,1,1,1,1,1,1] => 12
[5,1,1,1,1,1,1,1,1,1] => 4
[4,4,4,2] => 103
[4,4,4,1,1] => 103
[4,4,3,3] => 128
[4,4,3,2,1] => 397
[4,4,3,1,1,1] => 153
[4,4,2,2,2] => 47
[4,4,2,2,1,1] => 151
[4,4,2,1,1,1,1] => 67
[4,4,1,1,1,1,1,1] => 14
[4,3,3,3,1] => 115
[4,3,3,2,2] => 136
[4,3,3,2,1,1] => 254
[4,3,3,1,1,1,1] => 59
[4,3,2,2,2,1] => 136
[4,3,2,2,1,1,1] => 130
[4,3,2,1,1,1,1,1] => 57
[4,3,1,1,1,1,1,1,1] => 11
[4,2,2,2,2,2] => 6
[4,2,2,2,2,1,1] => 23
[4,2,2,2,1,1,1,1] => 19
[4,2,2,1,1,1,1,1,1] => 13
[4,2,1,1,1,1,1,1,1,1] => 6
[4,1,1,1,1,1,1,1,1,1,1] => 2
[3,3,3,3,2] => 20
[3,3,3,3,1,1] => 20
[3,3,3,2,2,1] => 62
[3,3,3,2,1,1,1] => 40
[3,3,3,1,1,1,1,1] => 8
[3,3,2,2,2,2] => 13
[3,3,2,2,2,1,1] => 42
[3,3,2,2,1,1,1,1] => 27
[3,3,2,1,1,1,1,1,1] => 13
[3,3,1,1,1,1,1,1,1,1] => 3
[3,2,2,2,2,2,1] => 8
[3,2,2,2,2,1,1,1] => 10
[3,2,2,2,1,1,1,1,1] => 9
[3,2,2,1,1,1,1,1,1,1] => 5
[3,2,1,1,1,1,1,1,1,1,1] => 3
[3,1,1,1,1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2,2,2] => 1
[2,2,2,2,2,2,1,1] => 1
[2,2,2,2,2,1,1,1,1] => 1
[2,2,2,2,1,1,1,1,1,1] => 1
[2,2,2,1,1,1,1,1,1,1,1] => 1
[2,2,1,1,1,1,1,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1
[15] => 19522
[14,1] => 20919
[13,2] => 11264
[13,1,1] => 11264
[12,3] => 8450
[12,2,1] => 12204
[12,1,1,1] => 3786
[11,4] => 6656
[11,3,1] => 9216
[11,2,2] => 3328
[11,2,1,1] => 6656
[11,1,1,1,1] => 1350
[10,5] => 5837
[10,4,1] => 7322
[10,3,2] => 5069
[10,3,1,1] => 5069
[10,2,2,1] => 3661
[10,2,1,1,1] => 2253
[10,1,1,1,1,1] => 420
[9,6] => 5420
[9,5,1] => 6486
[9,4,2] => 4068
[9,4,1,1] => 4068
[9,3,3] => 1948
[9,3,2,1] => 5632
[9,3,1,1,1] => 1736
[9,2,2,2] => 682
[9,2,2,1,1] => 2034
[9,2,1,1,1,1] => 780
[9,1,1,1,1,1,1] => 150
[8,7] => 5232
[8,6,1] => 6096
[8,5,2] => 3648
[8,5,1,1] => 3648
[8,4,3] => 3168
[8,4,2,1] => 4576
[8,4,1,1,1] => 1428
[8,3,3,1] => 2192
[8,3,2,2] => 1584
[8,3,2,1,1] => 3168
[8,3,1,1,1,1] => 615
[8,2,2,2,1] => 768
[8,2,2,1,1,1] => 704
[8,2,1,1,1,1,1] => 263
[8,1,1,1,1,1,1,1] => 56
[7,7,1] => 2991
[7,6,2] => 3485
[7,6,1,1] => 3485
[7,5,3] => 2888
[7,5,2,1] => 4171
[7,5,1,1,1] => 1298
[7,4,4] => 1308
[7,4,3,1] => 3622
[7,4,2,2] => 1308
[7,4,2,1,1] => 2616
[7,4,1,1,1,1] => 538
[7,3,3,2] => 1253
[7,3,3,1,1] => 1253
[7,3,2,2,1] => 1811
[7,3,2,1,1,1] => 1116
[7,3,1,1,1,1,1] => 200
[7,2,2,2,2] => 112
[7,2,2,2,1,1] => 439
[7,2,2,1,1,1,1] => 225
[7,2,1,1,1,1,1,1] => 95
[7,1,1,1,1,1,1,1,1] => 18
[6,6,3] => 1405
[6,6,2,1] => 2030
[6,6,1,1,1] => 630
[6,5,4] => 2430
[6,5,3,1] => 3365
[6,5,2,2] => 1215
[6,5,2,1,1] => 2430
[6,5,1,1,1,1] => 498
[6,4,4,1] => 1524
[6,4,3,2] => 2110
[6,4,3,1,1] => 2110
[6,4,2,2,1] => 1524
[6,4,2,1,1,1] => 938
[6,4,1,1,1,1,1] => 179
[6,3,3,3] => 335
[6,3,3,2,1] => 1460
[6,3,3,1,1,1] => 450
[6,3,2,2,2] => 355
[6,3,2,2,1,1] => 1055
[6,3,2,1,1,1,1] => 390
[6,3,1,1,1,1,1,1] => 76
[6,2,2,2,2,1] => 131
[6,2,2,2,1,1,1] => 157
[6,2,2,1,1,1,1,1] => 82
[6,2,1,1,1,1,1,1,1] => 36
[6,1,1,1,1,1,1,1,1,1] => 8
[5,5,5] => 389
[5,5,4,1] => 1454
[5,5,3,2] => 1007
[5,5,3,1,1] => 1007
[5,5,2,2,1] => 727
[5,5,2,1,1,1] => 447
[5,5,1,1,1,1,1] => 84
[5,4,4,2] => 912
[5,4,4,1,1] => 912
[5,4,3,3] => 874
[5,4,3,2,1] => 2526
[5,4,3,1,1,1] => 778
[5,4,2,2,2] => 306
[5,4,2,2,1,1] => 912
[5,4,2,1,1,1,1] => 354
[5,4,1,1,1,1,1,1] => 67
[5,3,3,3,1] => 402
[5,3,3,2,2] => 437
[5,3,3,2,1,1] => 874
[5,3,3,1,1,1,1] => 160
[5,3,2,2,2,1] => 424
[5,3,2,2,1,1,1] => 389
[5,3,2,1,1,1,1,1] => 140
[5,3,1,1,1,1,1,1,1] => 31
[5,2,2,2,2,2] => 16
[5,2,2,2,2,1,1] => 78
[5,2,2,2,1,1,1,1] => 50
[5,2,2,1,1,1,1,1,1] => 33
[5,2,1,1,1,1,1,1,1,1] => 12
[5,1,1,1,1,1,1,1,1,1,1] => 4
[4,4,4,3] => 262
[4,4,4,2,1] => 380
[4,4,4,1,1,1] => 124
[4,4,3,3,1] => 548
[4,4,3,2,2] => 396
[4,4,3,2,1,1] => 792
[4,4,3,1,1,1,1] => 158
[4,4,2,2,2,1] => 192
[4,4,2,2,1,1,1] => 176
[4,4,2,1,1,1,1,1] => 67
[4,4,1,1,1,1,1,1,1] => 14
[4,3,3,3,2] => 252
[4,3,3,3,1,1] => 252
[4,3,3,2,2,1] => 548
[4,3,3,2,1,1,1] => 338
[4,3,3,1,1,1,1,1] => 59
[4,3,2,2,2,2] => 68
[4,3,2,2,2,1,1] => 266
[4,3,2,2,1,1,1,1] => 131
[4,3,2,1,1,1,1,1,1] => 57
[4,3,1,1,1,1,1,1,1,1] => 11
[4,2,2,2,2,2,1] => 20
[4,2,2,2,2,1,1,1] => 30
[4,2,2,2,1,1,1,1,1] => 19
[4,2,2,1,1,1,1,1,1,1] => 13
[4,2,1,1,1,1,1,1,1,1,1] => 6
[4,1,1,1,1,1,1,1,1,1,1,1] => 2
[3,3,3,3,3] => 12
[3,3,3,3,2,1] => 87
[3,3,3,3,1,1,1] => 26
[3,3,3,2,2,2] => 43
[3,3,3,2,2,1,1] => 126
[3,3,3,2,1,1,1,1] => 40
[3,3,3,1,1,1,1,1,1] => 8
[3,3,2,2,2,2,1] => 47
[3,3,2,2,2,1,1,1] => 57
[3,3,2,2,1,1,1,1,1] => 27
[3,3,2,1,1,1,1,1,1,1] => 13
[3,3,1,1,1,1,1,1,1,1,1] => 3
[3,2,2,2,2,2,2] => 3
[3,2,2,2,2,2,1,1] => 14
[3,2,2,2,2,1,1,1,1] => 10
[3,2,2,2,1,1,1,1,1,1] => 9
[3,2,2,1,1,1,1,1,1,1,1] => 5
[3,2,1,1,1,1,1,1,1,1,1,1] => 3
[3,1,1,1,1,1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2,2,2,1] => 1
[2,2,2,2,2,2,1,1,1] => 1
[2,2,2,2,2,1,1,1,1,1] => 1
[2,2,2,2,1,1,1,1,1,1,1] => 1
[2,2,2,1,1,1,1,1,1,1,1,1] => 1
[2,2,1,1,1,1,1,1,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1
[16] => 72072
[15,1] => 76877
[14,2] => 42900
[14,1,1] => 42900
[13,3] => 30228
[13,2,1] => 46200
[13,1,1,1] => 15972
[12,4] => 24069
[12,3,1] => 32748
[12,2,2] => 12903
[12,2,1,1] => 24069
[12,1,1,1,1] => 4224
[11,5] => 20937
[11,4,1] => 26256
[11,3,2] => 18627
[11,3,1,1] => 18627
[11,2,2,1] => 14076
[11,2,1,1,1] => 9159
[11,1,1,1,1,1] => 1350
[10,6] => 19224
[10,5,1] => 23031
[10,4,2] => 15038
[10,4,1,1] => 15038
[10,3,3] => 6691
[10,3,2,1] => 20490
[10,3,1,1,1] => 7108
[10,2,2,2] => 2645
[10,2,2,1,1] => 7459
[10,2,1,1,1,1] => 2496
[10,1,1,1,1,1,1] => 420
[9,7] => 18306
[9,6,1] => 21360
[9,5,2] => 13326
[9,5,1,1] => 13326
[9,4,3] => 10936
[9,4,2,1] => 16709
[9,4,1,1,1] => 5773
[9,3,3,1] => 7434
[9,3,2,2] => 5874
[9,3,2,1,1] => 10936
[9,3,1,1,1,1] => 1942
[9,2,2,2,1] => 2939
[9,2,2,1,1,1] => 2711
[9,2,1,1,1,1,1] => 780
[9,1,1,1,1,1,1,1] => 150
[8,8] => 9010
[8,7,1] => 20595
[8,6,2] => 12515
[8,6,1,1] => 12515
[8,5,3] => 9812
[8,5,2,1] => 14992
[8,5,1,1,1] => 5180
[8,4,4] => 4521
[8,4,3,1] => 12304
[8,4,2,2] => 4846
[8,4,2,1,1] => 9042
[8,4,1,1,1,1] => 1584
[8,3,3,2] => 4366
[8,3,3,1,1] => 4366
[8,3,2,2,1] => 6608
[8,3,2,1,1,1] => 4308
[8,3,1,1,1,1,1] => 615
[8,2,2,2,2] => 419
[8,2,2,2,1,1] => 1594
[8,2,2,1,1,1,1] => 768
[8,2,1,1,1,1,1,1] => 263
[8,1,1,1,1,1,1,1,1] => 56
[7,7,2] => 6130
[7,7,1,1] => 6130
[7,6,3] => 9359
[7,6,2,1] => 14304
[7,6,1,1,1] => 4945
[7,5,4] => 8241
[7,5,3,1] => 11214
[7,5,2,2] => 4417
[7,5,2,1,1] => 8241
[7,5,1,1,1,1] => 1444
[7,4,4,1] => 5167
[7,4,3,2] => 7330
[7,4,3,1,1] => 7330
[7,4,2,2,1] => 5539
[7,4,2,1,1,1] => 3602
[7,4,1,1,1,1,1] => 538
[7,3,3,3] => 1084
[7,3,3,2,1] => 4990
[7,3,3,1,1,1] => 1737
[7,3,2,2,2] => 1291
[7,3,2,2,1,1] => 3635
[7,3,2,1,1,1,1] => 1229
[7,3,1,1,1,1,1,1] => 200
[7,2,2,2,2,1] => 479
[7,2,2,2,1,1,1] => 561
[7,2,2,1,1,1,1,1] => 225
[7,2,1,1,1,1,1,1,1] => 95
[7,1,1,1,1,1,1,1,1,1] => 18
[6,6,4] => 4013
[6,6,3,1] => 5460
[6,6,2,2] => 2151
[6,6,2,1,1] => 4013
[6,6,1,1,1,1] => 704
[6,5,5] => 3834
[6,5,4,1] => 9616
[6,5,3,2] => 6821
[6,5,3,1,1] => 6821
[6,5,2,2,1] => 5153
[6,5,2,1,1,1] => 3352
[6,5,1,1,1,1,1] => 498
[6,4,4,2] => 3138
[6,4,4,1,1] => 3138
[6,4,3,3] => 2794
[6,4,3,2,1] => 8552
[6,4,3,1,1,1] => 2964
[6,4,2,2,2] => 1104
[6,4,2,2,1,1] => 3114
[6,4,2,1,1,1,1] => 1040
[6,4,1,1,1,1,1,1] => 179
[6,3,3,3,1] => 1265
[6,3,3,2,2] => 1503
[6,3,3,2,1,1] => 2794
[6,3,3,1,1,1,1] => 501
[6,3,2,2,2,1] => 1506
[6,3,2,2,1,1,1] => 1390
[6,3,2,1,1,1,1,1] => 390
[6,3,1,1,1,1,1,1,1] => 76
[6,2,2,2,2,2] => 56
[6,2,2,2,2,1,1] => 269
[6,2,2,2,1,1,1,1] => 168
[6,2,2,1,1,1,1,1,1] => 82
[6,2,1,1,1,1,1,1,1,1] => 36
[6,1,1,1,1,1,1,1,1,1,1] => 8
[5,5,5,1] => 1531
[5,5,4,2] => 2998
[5,5,4,1,1] => 2998
[5,5,3,3] => 1335
[5,5,3,2,1] => 4086
[5,5,3,1,1,1] => 1416
[5,5,2,2,2] => 527
[5,5,2,2,1,1] => 1487
[5,5,2,1,1,1,1] => 496
[5,5,1,1,1,1,1,1] => 84
[5,4,4,3] => 2462
[5,4,4,2,1] => 3759
[5,4,4,1,1,1] => 1297
[5,4,3,3,1] => 3348
[5,4,3,2,2] => 2643
[5,4,3,2,1,1] => 4924
[5,4,3,1,1,1,1] => 871
[5,4,2,2,2,1] => 1322
[5,4,2,2,1,1,1] => 1219
[5,4,2,1,1,1,1,1] => 354
[5,4,1,1,1,1,1,1,1] => 67
[5,3,3,3,2] => 792
[5,3,3,3,1,1] => 792
[5,3,3,2,2,1] => 1801
[5,3,3,2,1,1,1] => 1176
[5,3,3,1,1,1,1,1] => 160
[5,3,2,2,2,2] => 229
[5,3,2,2,2,1,1] => 869
[5,3,2,2,1,1,1,1] => 422
[5,3,2,1,1,1,1,1,1] => 140
[5,3,1,1,1,1,1,1,1,1] => 31
[5,2,2,2,2,2,1] => 67
[5,2,2,2,2,1,1,1] => 94
[5,2,2,2,1,1,1,1,1] => 50
[5,2,2,1,1,1,1,1,1,1] => 33
[5,2,1,1,1,1,1,1,1,1,1] => 12
[5,1,1,1,1,1,1,1,1,1,1,1] => 4
[4,4,4,4] => 188
[4,4,4,3,1] => 1028
[4,4,4,2,2] => 405
[4,4,4,2,1,1] => 755
[4,4,4,1,1,1,1] => 132
[4,4,3,3,2] => 1095
[4,4,3,3,1,1] => 1095
[4,4,3,2,2,1] => 1657
[4,4,3,2,1,1,1] => 1079
[4,4,3,1,1,1,1,1] => 158
[4,4,2,2,2,2] => 105
[4,4,2,2,2,1,1] => 400
[4,4,2,2,1,1,1,1] => 192
[4,4,2,1,1,1,1,1,1] => 67
[4,4,1,1,1,1,1,1,1,1] => 14
[4,3,3,3,3] => 162
[4,3,3,3,2,1] => 993
[4,3,3,3,1,1,1] => 346
[4,3,3,2,2,2] => 386
[4,3,3,2,2,1,1] => 1086
[4,3,3,2,1,1,1,1] => 370
[4,3,3,1,1,1,1,1,1] => 59
[4,3,2,2,2,2,1] => 287
[4,3,2,2,2,1,1,1] => 336
[4,3,2,2,1,1,1,1,1] => 131
[4,3,2,1,1,1,1,1,1,1] => 57
[4,3,1,1,1,1,1,1,1,1,1] => 11
[4,2,2,2,2,2,2] => 7
[4,2,2,2,2,2,1,1] => 40
[4,2,2,2,2,1,1,1,1] => 31
[4,2,2,2,1,1,1,1,1,1] => 19
[4,2,2,1,1,1,1,1,1,1,1] => 13
[4,2,1,1,1,1,1,1,1,1,1,1] => 6
[4,1,1,1,1,1,1,1,1,1,1,1,1] => 2
[3,3,3,3,3,1] => 44
[3,3,3,3,2,2] => 87
[3,3,3,3,2,1,1] => 162
[3,3,3,3,1,1,1,1] => 29
[3,3,3,2,2,2,1] => 175
[3,3,3,2,2,1,1,1] => 162
[3,3,3,2,1,1,1,1,1] => 40
[3,3,3,1,1,1,1,1,1,1] => 8
[3,3,2,2,2,2,2] => 20
[3,3,2,2,2,2,1,1] => 94
[3,3,2,2,2,1,1,1,1] => 60
[3,3,2,2,1,1,1,1,1,1] => 27
[3,3,2,1,1,1,1,1,1,1,1] => 13
[3,3,1,1,1,1,1,1,1,1,1,1] => 3
[3,2,2,2,2,2,2,1] => 10
[3,2,2,2,2,2,1,1,1] => 15
[3,2,2,2,2,1,1,1,1,1] => 10
[3,2,2,2,1,1,1,1,1,1,1] => 9
[3,2,2,1,1,1,1,1,1,1,1,1] => 5
[3,2,1,1,1,1,1,1,1,1,1,1,1] => 3
[3,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2,2,2,2] => 1
[2,2,2,2,2,2,2,1,1] => 1
[2,2,2,2,2,2,1,1,1,1] => 1
[2,2,2,2,2,1,1,1,1,1,1] => 1
[2,2,2,2,1,1,1,1,1,1,1,1] => 1
[2,2,2,1,1,1,1,1,1,1,1,1,1] => 1
[2,2,1,1,1,1,1,1,1,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1
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Generating function
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Sequences for the coefficients of a few of the
first generating functions, in the case at hand:
6,1 6,2,3 5,3,2,2,2,1 6,1,5,2,0,2,1,2,1,0,0,1,1 6,3,2,1,1,2,2,2,0,2,2,0,1,0,1,2,0,1,0,0,0,0,0,1,0,0,1
$F_{1} = q$
$F_{2} = 2\ q$
$F_{3} = 3\ q$
$F_{4} = 5\ q$
$F_{5} = 6\ q + q^{2}$
$F_{6} = 6\ q + 2\ q^{2} + 3\ q^{3}$
$F_{7} = 5\ q + 3\ q^{2} + 2\ q^{3} + 2\ q^{4} + 2\ q^{5} + q^{6}$
$F_{8} = 6\ q + q^{2} + 5\ q^{3} + 2\ q^{4} + 2\ q^{6} + q^{7} + 2\ q^{8} + q^{9} + q^{12} + q^{13}$
$F_{9} = 6\ q + 3\ q^{2} + 2\ q^{3} + q^{4} + q^{5} + 2\ q^{6} + 2\ q^{7} + 2\ q^{8} + 2\ q^{10} + 2\ q^{11} + q^{13} + q^{15} + 2\ q^{16} + q^{18} + q^{24} + q^{27}$
$F_{10} = 7\ q + q^{2} + 2\ q^{3} + 2\ q^{4} + 2\ q^{5} + 3\ q^{6} + q^{8} + q^{11} + 6\ q^{12} + q^{15} + q^{16} + q^{18} + q^{19} + 2\ q^{27} + 2\ q^{32} + q^{34} + q^{38} + q^{39} + q^{45} + 2\ q^{48} + q^{55} + q^{77} + q^{86}$
$F_{11} = 7\ q + 2\ q^{2} + 2\ q^{3} + q^{4} + q^{5} + q^{6} + 3\ q^{8} + q^{9} + 2\ q^{11} + q^{12} + 2\ q^{13} + 2\ q^{14} + q^{18} + q^{19} + q^{25} + q^{27} + q^{28} + q^{30} + q^{31} + q^{34} + q^{36} + q^{39} + 2\ q^{46} + q^{50} + q^{54} + q^{56} + 2\ q^{62} + 2\ q^{71} + q^{77} + 2\ q^{83} + q^{85} + q^{97} + q^{99} + q^{113} + 2\ q^{138} + q^{155} + q^{210} + q^{231}$
$F_{12} = 8\ q + q^{2} + 2\ q^{3} + 2\ q^{4} + 2\ q^{5} + q^{6} + q^{7} + 2\ q^{8} + 2\ q^{9} + q^{11} + q^{12} + 2\ q^{13} + q^{14} + q^{18} + q^{19} + q^{20} + q^{24} + 2\ q^{27} + 2\ q^{31} + q^{32} + q^{33} + q^{36} + q^{55} + 2\ q^{56} + q^{57} + q^{61} + q^{62} + q^{67} + q^{72} + 2\ q^{76} + 3\ q^{78} + q^{84} + q^{87} + q^{95} + q^{109} + q^{113} + q^{130} + 2\ q^{132} + q^{150} + q^{163} + q^{164} + 2\ q^{185} + q^{196} + q^{216} + q^{220} + 2\ q^{246} + q^{257} + q^{280} + q^{288} + q^{324} + q^{384} + q^{386} + q^{428} + q^{640} + q^{700}$
$F_{13} = 8\ q + q^{2} + 3\ q^{3} + q^{4} + q^{5} + q^{6} + 2\ q^{8} + q^{9} + q^{10} + q^{11} + 2\ q^{12} + 2\ q^{13} + 2\ q^{14} + q^{17} + 2\ q^{18} + q^{19} + q^{27} + q^{29} + q^{31} + q^{33} + q^{36} + 2\ q^{37} + q^{43} + q^{46} + q^{56} + q^{57} + 2\ q^{59} + 2\ q^{67} + q^{74} + q^{76} + q^{82} + q^{84} + q^{87} + q^{90} + q^{95} + q^{116} + 2\ q^{120} + q^{125} + q^{128} + q^{137} + q^{140} + q^{142} + q^{146} + q^{150} + q^{156} + q^{161} + q^{179} + q^{180} + q^{187} + q^{194} + q^{200} + q^{210} + q^{222} + q^{263} + q^{273} + q^{297} + 3\ q^{308} + q^{347} + q^{365} + 2\ q^{368} + 2\ q^{399} + q^{420} + q^{465} + 2\ q^{466} + q^{513} + q^{533} + q^{536} + 2\ q^{601} + q^{613} + q^{675} + q^{730} + q^{811} + 2\ q^{1000} + q^{1098} + q^{1650} + q^{1790}$
$F_{14} = 9\ q + q^{2} + 2\ q^{3} + q^{4} + q^{5} + 2\ q^{6} + 3\ q^{8} + q^{9} + q^{10} + q^{11} + q^{12} + 3\ q^{13} + q^{14} + q^{18} + q^{19} + 2\ q^{20} + q^{23} + q^{27} + q^{31} + q^{33} + q^{36} + q^{37} + q^{40} + q^{42} + q^{44} + q^{47} + q^{50} + q^{56} + q^{57} + q^{59} + q^{62} + 2\ q^{67} + q^{76} + q^{82} + q^{84} + q^{92} + q^{95} + 2\ q^{103} + q^{107} + q^{115} + q^{128} + q^{130} + q^{132} + 2\ q^{136} + q^{140} + q^{150} + q^{151} + q^{153} + q^{156} + q^{179} + q^{200} + q^{201} + q^{204} + q^{221} + q^{229} + q^{254} + q^{263} + q^{309} + q^{344} + q^{346} + q^{352} + 2\ q^{355} + q^{371} + q^{380} + q^{397} + q^{412} + q^{420} + q^{422} + q^{440} + q^{463} + q^{470} + q^{478} + q^{494} + q^{504} + q^{507} + q^{515} + q^{517} + q^{538} + q^{576} + q^{577} + q^{594} + 2\ q^{631} + q^{718} + q^{738} + q^{756} + q^{783} + 2\ q^{844} + q^{966} + q^{983} + 2\ q^{1071} + q^{1080} + 2\ q^{1179} + q^{1244} + q^{1296} + q^{1350} + 2\ q^{1386} + q^{1452} + q^{1548} + q^{1560} + q^{1661} + q^{1743} + 2\ q^{1753} + q^{1944} + q^{2140} + q^{2354} + 2\ q^{3126} + q^{3402} + q^{4973} + q^{5346}$
$F_{15} = 9\ q + q^{2} + 3\ q^{3} + q^{4} + q^{5} + q^{6} + 2\ q^{8} + q^{9} + q^{10} + q^{11} + 2\ q^{12} + 2\ q^{13} + 2\ q^{14} + q^{16} + q^{18} + q^{19} + q^{20} + q^{26} + q^{27} + q^{30} + q^{31} + q^{33} + q^{36} + q^{40} + q^{43} + q^{47} + q^{50} + q^{56} + 2\ q^{57} + q^{59} + 2\ q^{67} + q^{68} + q^{76} + q^{78} + q^{82} + q^{84} + q^{87} + q^{95} + q^{112} + q^{124} + q^{126} + 2\ q^{131} + q^{140} + q^{150} + q^{157} + q^{158} + q^{160} + q^{176} + q^{179} + q^{192} + q^{200} + q^{225} + 2\ q^{252} + q^{262} + q^{263} + q^{266} + q^{306} + q^{335} + q^{338} + q^{354} + q^{355} + q^{380} + 2\ q^{389} + q^{390} + q^{396} + q^{402} + q^{420} + q^{424} + q^{437} + q^{439} + q^{447} + q^{450} + q^{498} + q^{538} + 2\ q^{548} + q^{615} + q^{630} + q^{682} + q^{704} + q^{727} + q^{768} + q^{778} + q^{780} + q^{792} + 2\ q^{874} + 3\ q^{912} + q^{938} + 2\ q^{1007} + q^{1055} + q^{1116} + q^{1215} + 2\ q^{1253} + q^{1298} + 2\ q^{1308} + q^{1350} + q^{1405} + q^{1428} + q^{1454} + q^{1460} + 2\ q^{1524} + q^{1584} + q^{1736} + q^{1811} + q^{1948} + q^{2030} + q^{2034} + 2\ q^{2110} + q^{2192} + q^{2253} + 2\ q^{2430} + q^{2526} + q^{2616} + q^{2888} + q^{2991} + 2\ q^{3168} + q^{3328} + q^{3365} + 2\ q^{3485} + q^{3622} + 2\ q^{3648} + q^{3661} + q^{3786} + 2\ q^{4068} + q^{4171} + q^{4576} + 2\ q^{5069} + q^{5232} + q^{5420} + q^{5632} + q^{5837} + q^{6096} + q^{6486} + 2\ q^{6656} + q^{7322} + q^{8450} + q^{9216} + 2\ q^{11264} + q^{12204} + q^{19522} + q^{20919}$
Description
The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition.
References
[1] Schocker, M. Multiplicities of higher Lie characters MathSciNet:1984625
Code
def statistic(la):
s = SymmetricFunctions(ZZ).s()
return max(c for _, c in s.higher_lie_character(la))
Created
May 25, 2023 at 14:39 by Martin Rubey
Updated
May 25, 2023 at 14:39 by Martin Rubey
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