Identifier
Identifier
Values
[1] => 1
[1,1] => 1
[2] => 1
[1,1,1] => 1
[1,2] => 2
[2,1] => 2
[3] => 1
[1,1,1,1] => 1
[1,1,2] => 3
[1,2,1] => 5
[1,3] => 3
[2,1,1] => 3
[2,2] => 5
[3,1] => 3
[4] => 1
[1,1,1,1,1] => 1
[1,1,1,2] => 4
[1,1,2,1] => 9
[1,1,3] => 6
[1,2,1,1] => 9
[1,2,2] => 16
[1,3,1] => 11
[1,4] => 4
[2,1,1,1] => 4
[2,1,2] => 11
[2,2,1] => 16
[2,3] => 9
[3,1,1] => 6
[3,2] => 9
[4,1] => 4
[5] => 1
[1,1,1,1,1,1] => 1
[1,1,1,1,2] => 5
[1,1,1,2,1] => 14
[1,1,1,3] => 10
[1,1,2,1,1] => 19
[1,1,2,2] => 35
[1,1,3,1] => 26
[1,1,4] => 10
[1,2,1,1,1] => 14
[1,2,1,2] => 40
[1,2,2,1] => 61
[1,2,3] => 35
[1,3,1,1] => 26
[1,3,2] => 40
[1,4,1] => 19
[1,5] => 5
[2,1,1,1,1] => 5
[2,1,1,2] => 19
[2,1,2,1] => 40
[2,1,3] => 26
[2,2,1,1] => 35
[2,2,2] => 61
[2,3,1] => 40
[2,4] => 14
[3,1,1,1] => 10
[3,1,2] => 26
[3,2,1] => 35
[3,3] => 19
[4,1,1] => 10
[4,2] => 14
[5,1] => 5
[6] => 1
[1,1,1,1,1,1,1] => 1
[1,1,1,1,1,2] => 6
[1,1,1,1,2,1] => 20
[1,1,1,1,3] => 15
[1,1,1,2,1,1] => 34
[1,1,1,2,2] => 64
[1,1,1,3,1] => 50
[1,1,1,4] => 20
[1,1,2,1,1,1] => 34
[1,1,2,1,2] => 99
[1,1,2,2,1] => 155
[1,1,2,3] => 90
[1,1,3,1,1] => 71
[1,1,3,2] => 111
[1,1,4,1] => 55
[1,1,5] => 15
[1,2,1,1,1,1] => 20
[1,2,1,1,2] => 78
[1,2,1,2,1] => 169
[1,2,1,3] => 111
[1,2,2,1,1] => 155
[1,2,2,2] => 272
[1,2,3,1] => 181
[1,2,4] => 64
[1,3,1,1,1] => 50
[1,3,1,2] => 132
[1,3,2,1] => 181
[1,3,3] => 99
[1,4,1,1] => 55
[1,4,2] => 78
[1,5,1] => 29
[1,6] => 6
[2,1,1,1,1,1] => 6
[2,1,1,1,2] => 29
[2,1,1,2,1] => 78
[2,1,1,3] => 55
[2,1,2,1,1] => 99
[2,1,2,2] => 181
[2,1,3,1] => 132
[2,1,4] => 50
[2,2,1,1,1] => 64
[2,2,1,2] => 181
[2,2,2,1] => 272
[2,2,3] => 155
[2,3,1,1] => 111
[2,3,2] => 169
[2,4,1] => 78
[2,5] => 20
[3,1,1,1,1] => 15
[3,1,1,2] => 55
[3,1,2,1] => 111
[3,1,3] => 71
[3,2,1,1] => 90
[3,2,2] => 155
[3,3,1] => 99
[3,4] => 34
[4,1,1,1] => 20
[4,1,2] => 50
[4,2,1] => 64
[4,3] => 34
[5,1,1] => 15
[5,2] => 20
[6,1] => 6
[7] => 1
[1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,2] => 7
[1,1,1,1,1,2,1] => 27
[1,1,1,1,1,3] => 21
[1,1,1,1,2,1,1] => 55
[1,1,1,1,2,2] => 105
[1,1,1,1,3,1] => 85
[1,1,1,1,4] => 35
[1,1,1,2,1,1,1] => 69
[1,1,1,2,1,2] => 203
[1,1,1,2,2,1] => 323
[1,1,1,2,3] => 189
[1,1,1,3,1,1] => 155
[1,1,1,3,2] => 245
[1,1,1,4,1] => 125
[1,1,1,5] => 35
[1,1,2,1,1,1,1] => 55
[1,1,2,1,1,2] => 217
[1,1,2,1,2,1] => 477
[1,1,2,1,3] => 315
[1,1,2,2,1,1] => 449
[1,1,2,2,2] => 791
[1,1,2,3,1] => 531
[1,1,2,4] => 189
[1,1,3,1,1,1] => 155
[1,1,3,1,2] => 413
[1,1,3,2,1] => 573
[1,1,3,3] => 315
[1,1,4,1,1] => 181
[1,1,4,2] => 259
[1,1,5,1] => 99
[1,1,6] => 21
[1,2,1,1,1,1,1] => 27
[1,2,1,1,1,2] => 133
[1,2,1,1,2,1] => 365
[1,2,1,1,3] => 259
[1,2,1,2,1,1] => 477
[1,2,1,2,2] => 875
[1,2,1,3,1] => 643
[1,2,1,4] => 245
[1,2,2,1,1,1] => 323
[1,2,2,1,2] => 917
[1,2,2,2,1] => 1385
[1,2,2,3] => 791
[1,2,3,1,1] => 573
[1,2,3,2] => 875
[1,2,4,1] => 407
[1,2,5] => 105
[1,3,1,1,1,1] => 85
[1,3,1,1,2] => 315
[1,3,1,2,1] => 643
[1,3,1,3] => 413
[1,3,2,1,1] => 531
[1,3,2,2] => 917
[1,3,3,1] => 589
[1,3,4] => 203
[1,4,1,1,1] => 125
[1,4,1,2] => 315
[1,4,2,1] => 407
[1,4,3] => 217
[1,5,1,1] => 99
[1,5,2] => 133
[1,6,1] => 41
[1,7] => 7
[2,1,1,1,1,1,1] => 7
[2,1,1,1,1,2] => 41
[2,1,1,1,2,1] => 133
[2,1,1,1,3] => 99
[2,1,1,2,1,1] => 217
[2,1,1,2,2] => 407
[2,1,1,3,1] => 315
[2,1,1,4] => 125
[2,1,2,1,1,1] => 203
[2,1,2,1,2] => 589
[2,1,2,2,1] => 917
[2,1,2,3] => 531
[2,1,3,1,1] => 413
[2,1,3,2] => 643
[2,1,4,1] => 315
[2,1,5] => 85
[2,2,1,1,1,1] => 105
[2,2,1,1,2] => 407
[2,2,1,2,1] => 875
[2,2,1,3] => 573
[2,2,2,1,1] => 791
[2,2,2,2] => 1385
[2,2,3,1] => 917
[2,2,4] => 323
[2,3,1,1,1] => 245
[2,3,1,2] => 643
[2,3,2,1] => 875
[2,3,3] => 477
[2,4,1,1] => 259
[2,4,2] => 365
[2,5,1] => 133
[2,6] => 27
[3,1,1,1,1,1] => 21
[3,1,1,1,2] => 99
[3,1,1,2,1] => 259
[3,1,1,3] => 181
[3,1,2,1,1] => 315
[3,1,2,2] => 573
[3,1,3,1] => 413
[3,1,4] => 155
[3,2,1,1,1] => 189
[3,2,1,2] => 531
[3,2,2,1] => 791
[3,2,3] => 449
[3,3,1,1] => 315
[3,3,2] => 477
[3,4,1] => 217
[3,5] => 55
[4,1,1,1,1] => 35
[4,1,1,2] => 125
[4,1,2,1] => 245
[4,1,3] => 155
[4,2,1,1] => 189
[4,2,2] => 323
[4,3,1] => 203
[4,4] => 69
[5,1,1,1] => 35
[5,1,2] => 85
[5,2,1] => 105
[5,3] => 55
[6,1,1] => 21
[6,2] => 27
[7,1] => 7
[8] => 1
[1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,2] => 8
[1,1,1,1,1,1,2,1] => 35
[1,1,1,1,1,1,3] => 28
[1,1,1,1,1,2,1,1] => 83
[1,1,1,1,1,2,2] => 160
[1,1,1,1,1,3,1] => 133
[1,1,1,1,1,4] => 56
[1,1,1,1,2,1,1,1] => 125
[1,1,1,1,2,1,2] => 370
[1,1,1,1,2,2,1] => 595
[1,1,1,1,2,3] => 350
[1,1,1,1,3,1,1] => 295
[1,1,1,1,3,2] => 470
[1,1,1,1,4,1] => 245
[1,1,1,1,5] => 70
[1,1,1,2,1,1,1,1] => 125
[1,1,1,2,1,1,2] => 496
[1,1,1,2,1,2,1] => 1099
[1,1,1,2,1,3] => 728
[1,1,1,2,2,1,1] => 1051
[1,1,1,2,2,2] => 1856
[1,1,1,2,3,1] => 1253
[1,1,1,2,4] => 448
[1,1,1,3,1,1,1] => 379
[1,1,1,3,1,2] => 1016
[1,1,1,3,2,1] => 1421
[1,1,1,3,3] => 784
[1,1,1,4,1,1] => 461
[1,1,1,4,2] => 664
[1,1,1,5,1] => 259
[1,1,1,6] => 56
[1,1,2,1,1,1,1,1] => 83
[1,1,2,1,1,1,2] => 412
[1,1,2,1,1,2,1] => 1141
[1,1,2,1,1,3] => 812
[1,1,2,1,2,1,1] => 1513
[1,1,2,1,2,2] => 2780
[1,1,2,1,3,1] => 2051
[1,1,2,1,4] => 784
[1,1,2,2,1,1,1] => 1051
[1,1,2,2,1,2] => 2990
[1,1,2,2,2,1] => 4529
[1,1,2,2,3] => 2590
[1,1,2,3,1,1] => 1889
[1,1,2,3,2] => 2890
[1,1,2,4,1] => 1351
[1,1,2,5] => 350
[1,1,3,1,1,1,1] => 295
[1,1,3,1,1,2] => 1100
[1,1,3,1,2,1] => 2261
[1,1,3,1,3] => 1456
[1,1,3,2,1,1] => 1889
[1,1,3,2,2] => 3268
[1,1,3,3,1] => 2107
[1,1,3,4] => 728
[1,1,4,1,1,1] => 461
[1,1,4,1,2] => 1168
[1,1,4,2,1] => 1519
[1,1,4,3] => 812
[1,1,5,1,1] => 379
[1,1,5,2] => 512
[1,1,6,1] => 161
[1,1,7] => 28
[1,2,1,1,1,1,1,1] => 35
[1,2,1,1,1,1,2] => 208
[1,2,1,1,1,2,1] => 685
[1,2,1,1,1,3] => 512
[1,2,1,1,2,1,1] => 1141
[1,2,1,1,2,2] => 2144
[1,2,1,1,3,1] => 1667
[1,2,1,1,4] => 664
[1,2,1,2,1,1,1] => 1099
[1,2,1,2,1,2] => 3194
[1,2,1,2,2,1] => 4985
[1,2,1,2,3] => 2890
[1,2,1,3,1,1] => 2261
[1,2,1,3,2] => 3526
[1,2,1,4,1] => 1735
[1,2,1,5] => 470
[1,2,2,1,1,1,1] => 595
[1,2,2,1,1,2] => 2312
[1,2,2,1,2,1] => 4985
[1,2,2,1,3] => 3268
[1,2,2,2,1,1] => 4529
[1,2,2,2,2] => 7936
[1,2,2,3,1] => 5263
[1,2,2,4] => 1856
[1,2,3,1,1,1] => 1421
[1,2,3,1,2] => 3736
[1,2,3,2,1] => 5095
[1,2,3,3] => 2780
[1,2,4,1,1] => 1519
[1,2,4,2] => 2144
[1,2,5,1] => 785
[1,2,6] => 160
[1,3,1,1,1,1,1] => 133
[1,3,1,1,1,2] => 632
[1,3,1,1,2,1] => 1667
[1,3,1,1,3] => 1168
[1,3,1,2,1,1] => 2051
[1,3,1,2,2] => 3736
[1,3,1,3,1] => 2701
[1,3,1,4] => 1016
[1,3,2,1,1,1] => 1253
[1,3,2,1,2] => 3526
[1,3,2,2,1] => 5263
[1,3,2,3] => 2990
[1,3,3,1,1] => 2107
[1,3,3,2] => 3194
[1,3,4,1] => 1457
[1,3,5] => 370
[1,4,1,1,1,1] => 245
[1,4,1,1,2] => 880
[1,4,1,2,1] => 1735
[1,4,1,3] => 1100
[1,4,2,1,1] => 1351
[1,4,2,2] => 2312
[1,4,3,1] => 1457
[1,4,4] => 496
[1,5,1,1,1] => 259
[1,5,1,2] => 632
[1,5,2,1] => 785
[1,5,3] => 412
[1,6,1,1] => 161
[1,6,2] => 208
[1,7,1] => 55
[1,8] => 8
[2,1,1,1,1,1,1,1] => 8
[2,1,1,1,1,1,2] => 55
[2,1,1,1,1,2,1] => 208
[2,1,1,1,1,3] => 161
[2,1,1,1,2,1,1] => 412
[2,1,1,1,2,2] => 785
[2,1,1,1,3,1] => 632
[2,1,1,1,4] => 259
[2,1,1,2,1,1,1] => 496
[2,1,1,2,1,2] => 1457
[2,1,1,2,2,1] => 2312
[2,1,1,2,3] => 1351
[2,1,1,3,1,1] => 1100
[2,1,1,3,2] => 1735
[2,1,1,4,1] => 880
[2,1,1,5] => 245
[2,1,2,1,1,1,1] => 370
[2,1,2,1,1,2] => 1457
[2,1,2,1,2,1] => 3194
[2,1,2,1,3] => 2107
[2,1,2,2,1,1] => 2990
[2,1,2,2,2] => 5263
[2,1,2,3,1] => 3526
[2,1,2,4] => 1253
[2,1,3,1,1,1] => 1016
[2,1,3,1,2] => 2701
[2,1,3,2,1] => 3736
[2,1,3,3] => 2051
[2,1,4,1,1] => 1168
[2,1,4,2] => 1667
[2,1,5,1] => 632
[2,1,6] => 133
[2,2,1,1,1,1,1] => 160
[2,2,1,1,1,2] => 785
[2,2,1,1,2,1] => 2144
[2,2,1,1,3] => 1519
[2,2,1,2,1,1] => 2780
[2,2,1,2,2] => 5095
[2,2,1,3,1] => 3736
[2,2,1,4] => 1421
[2,2,2,1,1,1] => 1856
[2,2,2,1,2] => 5263
[2,2,2,2,1] => 7936
[2,2,2,3] => 4529
[2,2,3,1,1] => 3268
[2,2,3,2] => 4985
[2,2,4,1] => 2312
[2,2,5] => 595
[2,3,1,1,1,1] => 470
[2,3,1,1,2] => 1735
[2,3,1,2,1] => 3526
[2,3,1,3] => 2261
[2,3,2,1,1] => 2890
[2,3,2,2] => 4985
[2,3,3,1] => 3194
[2,3,4] => 1099
[2,4,1,1,1] => 664
[2,4,1,2] => 1667
[2,4,2,1] => 2144
[2,4,3] => 1141
[2,5,1,1] => 512
[2,5,2] => 685
[2,6,1] => 208
[2,7] => 35
[3,1,1,1,1,1,1] => 28
[3,1,1,1,1,2] => 161
[3,1,1,1,2,1] => 512
[3,1,1,1,3] => 379
[3,1,1,2,1,1] => 812
[3,1,1,2,2] => 1519
[3,1,1,3,1] => 1168
[3,1,1,4] => 461
[3,1,2,1,1,1] => 728
[3,1,2,1,2] => 2107
[3,1,2,2,1] => 3268
[3,1,2,3] => 1889
[3,1,3,1,1] => 1456
[3,1,3,2] => 2261
[3,1,4,1] => 1100
[3,1,5] => 295
[3,2,1,1,1,1] => 350
[3,2,1,1,2] => 1351
[3,2,1,2,1] => 2890
[3,2,1,3] => 1889
[3,2,2,1,1] => 2590
[3,2,2,2] => 4529
[3,2,3,1] => 2990
[3,2,4] => 1051
[3,3,1,1,1] => 784
[3,3,1,2] => 2051
[3,3,2,1] => 2780
[3,3,3] => 1513
[3,4,1,1] => 812
[3,4,2] => 1141
[3,5,1] => 412
[3,6] => 83
[4,1,1,1,1,1] => 56
[4,1,1,1,2] => 259
[4,1,1,2,1] => 664
[4,1,1,3] => 461
[4,1,2,1,1] => 784
[4,1,2,2] => 1421
[4,1,3,1] => 1016
[4,1,4] => 379
[4,2,1,1,1] => 448
[4,2,1,2] => 1253
[4,2,2,1] => 1856
[4,2,3] => 1051
[4,3,1,1] => 728
[4,3,2] => 1099
[4,4,1] => 496
[4,5] => 125
[5,1,1,1,1] => 70
[5,1,1,2] => 245
[5,1,2,1] => 470
[5,1,3] => 295
[5,2,1,1] => 350
[5,2,2] => 595
[5,3,1] => 370
[5,4] => 125
[6,1,1,1] => 56
[6,1,2] => 133
[6,2,1] => 160
[6,3] => 83
[7,1,1] => 28
[7,2] => 35
[8,1] => 8
[9] => 1
[1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,2,2] => 231
[1,1,1,1,2,1,1,2] => 999
[1,1,1,1,2,2,1,1] => 2149
[1,1,1,1,3,3] => 1674
[1,1,1,1,5,1] => 574
[1,1,2,1,1,1,1,2] => 711
[1,1,2,1,1,2,1,1] => 3961
[1,1,2,1,2,3] => 10206
[1,1,2,2,1,1,1,1] => 2149
[1,1,2,2,2,2] => 28839
[1,1,3,1,1,3] => 4536
[1,1,3,2,1,2] => 13941
[1,1,3,3,1,1] => 8371
[1,1,4,4] => 2064
[1,1,7,1] => 244
[1,2,1,1,4,1] => 5191
[1,2,6,1] => 1369
[1,3,1,1,3,1] => 8371
[1,3,5,1] => 3079
[1,4,1,1,2,1] => 5191
[1,4,4,1] => 3961
[1,5,1,1,1,1] => 574
[1,5,3,1] => 3079
[1,6,1,2] => 1134
[1,6,2,1] => 1369
[1,7,1,1] => 244
[1,8,1] => 71
[1,9] => 9
[2,1,1,1,1,1,1,2] => 71
[2,1,1,1,1,2,1,1] => 711
[2,1,1,1,2,3] => 2906
[2,1,1,2,1,1,1,1] => 999
[2,1,1,2,2,2] => 14759
[2,1,2,1,1,3] => 6056
[2,1,2,2,1,2] => 22121
[2,1,2,3,1,1] => 13941
[2,1,3,4] => 5264
[2,1,6,1] => 1134
[2,2,1,1,1,1,1,1] => 231
[2,2,1,1,2,2] => 13991
[2,2,2,1,1,2] => 14759
[2,2,2,2,1,1] => 28839
[2,2,3,3] => 17594
[3,1,1,1,1,3] => 701
[3,1,1,2,1,2] => 6056
[3,1,1,3,1,1] => 4536
[3,1,2,4] => 4949
[3,2,1,1,1,2] => 2906
[3,2,1,2,1,1] => 10206
[3,2,2,3] => 16451
[3,3,1,1,1,1] => 1674
[3,3,2,2] => 17594
[4,1,1,4] => 1301
[4,2,1,3] => 4949
[4,3,1,2] => 5264
[4,4,1,1] => 2064
[5,5] => 251
[8,1,1] => 36
[9,1] => 9
[10,1] => 10
[1,10] => 10
[1,8,1,1] => 351
[1,7,2,1] => 2221
[1,6,3,1] => 5851
[1,5,4,1] => 9151
[1,4,5,1] => 9151
[1,3,6,1] => 5851
[1,2,7,1] => 2221
[1,1,8,1] => 351
[5,5,1,1] => 11825
[1,1,5,5] => 11825
[1,1,4,1,1,4] => 55220
[4,1,1,4,1,1] => 55220
[1,1,3,1,2,4] => 185108
[1,1,4,2,1,3] => 215160
[3,1,2,4,1,1] => 215160
[4,2,1,3,1,1] => 185108
[1,1,2,1,3,4] => 150227
[1,1,3,1,1,1,1,3] => 25080
[1,1,4,3,1,2] => 230241
[2,2,4,4] => 157475
[2,1,3,4,1,1] => 230241
[3,1,1,1,1,3,1,1] => 25080
[4,4,2,2] => 157475
[4,3,1,2,1,1] => 150227
[1,1,1,1,4,4] => 16995
[1,1,2,1,2,1,1,3] => 169455
[1,1,3,2,2,3] => 635745
[1,1,3,1,1,2,1,2] => 221529
[1,1,4,4,1,1] => 90509
[2,2,3,1,1,3] => 385395
[2,1,2,1,1,3,1,1] => 221529
[3,3,3,3] => 315523
[3,2,2,3,1,1] => 635745
[3,1,1,3,2,2] => 385395
[3,1,1,2,1,2,1,1] => 169455
[4,4,1,1,1,1] => 16995
[1,1,1,1,3,1,1,3] => 34375
[1,1,2,1,1,1,2,3] => 75570
[1,1,2,1,2,2,1,2] => 622314
[1,1,3,2,1,1,1,2] => 111474
[1,1,3,1,1,3,1,1] => 166541
[2,2,2,1,2,3] => 996390
[2,2,3,2,1,2] => 1152546
[2,1,1,2,3,3] => 280774
[2,1,1,1,2,3,1,1] => 111474
[2,1,2,3,2,2] => 1152546
[2,1,2,2,1,2,1,1] => 622314
[3,3,2,1,1,2] => 280774
[3,2,1,2,2,2] => 996390
[3,2,1,1,1,2,1,1] => 75570
[3,1,1,3,1,1,1,1] => 34375
[1,1,1,1,1,1,2,1,1,2] => 3157
[1,1,1,1,1,1,1,1,2,2] => 429
[2,2,1,1,2,1,1,2] => 199341
[2,2,1,1,1,1,2,2] => 66989
[2,1,1,2,2,1,1,2] => 228205
[1,1,2,2,2,1,1,2] => 450021
[2,1,1,2,1,1,2,2] => 199341
[1,1,2,2,1,1,2,2] => 421421
[1,1,1,1,3,2,1,2] => 107866
[1,1,2,1,1,1,1,1,1,2] => 1749
[1,1,2,1,1,2,2,2] => 396077
[3,3,1,1,2,2] => 312102
[1,1,1,1,2,1,1,1,1,2] => 4741
[1,1,1,1,2,2,2,2] => 203181
[3,2,2,2,1,2] => 1022845
[2,1,1,1,2,2,1,2] => 174514
[1,1,1,1,2,1,2,3] => 69850
[3,2,1,1,1,1,1,2] => 9910
[2,1,1,1,1,1,1,1,1,2] => 109
[2,1,1,1,1,2,2,2] => 72621
[2,2,2,1,1,1,1,2] => 72621
[2,2,2,2,2,2] => 2702765
[1,1,1,1,1,1,3,3] => 5698
[3,1,2,3,1,2] => 527284
[2,1,3,3,1,2] => 578665
[3,1,1,1,1,2,1,2] => 30700
[2,1,2,1,1,2,1,2] => 291169
[3,1,1,2,1,1,1,2] => 45715
[2,1,2,2,1,1,1,2] => 174514
[1,1,3,3,2,2] => 686906
[2,2,1,1,3,3] => 312102
[3,3,1,1,1,1,1,1] => 5698
[2,1,2,3,1,1,1,1] => 107866
[1,1,3,3,1,1,1,1] => 65154
[2,2,3,3,1,1] => 686906
[1,1,1,1,3,3,1,1] => 65154
[3,1,1,1,1,1,1,3] => 1891
[3,1,1,2,2,3] => 294106
[3,2,1,1,2,3] => 251371
[2,1,1,1,1,1,2,3] => 9910
[1,1,2,1,2,3,1,1] => 392822
[4,2,1,1,1,3] => 43251
[3,2,2,1,1,3] => 294106
[4,2,1,2,1,2] => 262932
[2,1,2,2,2,3] => 1022845
[4,3,1,1,1,2] => 46947
[1,1,2,2,3,3] => 538890
[3,3,2,2,1,1] => 538890
[1,1,2,2,2,2,1,1] => 880529
[2,2,1,1,1,1,1,1,1,1] => 429
[2,2,1,1,2,2,1,1] => 421421
[2,1,1,1,2,1,1,3] => 45715
[2,2,2,2,1,1,1,1] => 203181
[2,1,1,2,1,1,1,1,1,1] => 3157
[2,1,1,1,3,4] => 46947
[2,1,1,1,1,1,1,2,1,1] => 1749
[2,1,2,1,1,1,1,3] => 30700
[2,2,2,1,1,2,1,1] => 396077
[2,1,1,1,1,2,1,1,1,1] => 4741
[2,1,2,1,2,4] => 262932
[2,1,1,2,2,2,1,1] => 450021
[2,1,3,2,1,3] => 527284
[2,1,3,1,1,4] => 123540
[3,1,1,1,2,4] => 43251
[1,1,2,1,1,2,1,1,1,1] => 26709
[1,1,1,1,2,1,1,2,1,1] => 26709
[1,1,2,2,1,1,1,1,1,1] => 6909
[1,1,1,1,2,2,1,1,1,1] => 15049
[1,1,1,1,1,1,2,2,1,1] => 6909
[1,1,1,1,1,1,1,1,1,1,1,1] => 1
[5,4,1,2] => 39225
[4,1,1,3,1,2] => 123540
[4,3,2,3] => 215346
[5,3,1,3] => 54966
[4,1,1,2,1,3] => 90771
[3,2,1,2,1,1,1,1] => 69850
[1,1,3,2,1,2,1,1] => 392822
[5,2,1,4] => 36926
[5,1,1,5] => 8051
[3,1,2,2,1,3] => 458083
[4,2,2,4] => 147443
[3,1,2,1,1,4] => 90771
[4,1,1,1,1,4] => 8051
[3,1,3,5] => 54966
[6,6] => 923
[4,1,2,5] => 36926
[2,1,4,5] => 39225
[3,2,3,4] => 215346
[1,1,2,1,1,1,1,2,1,1] => 18041
click to show generating function       
Description
The number of ribbon shaped standard tableaux.
A ribbon is a connected skew shape which does not contain a $2\times 2$ square. The set of ribbon shapes are therefore in bijection with integer compositons, the parts of the composition specify the row lengths. This statistic records the number of standard tableaux of the given shape.
This is also the size of the preimage of the map 'descent composition' Mp00071descent composition from permutations to integer compositions: reading a tableau from bottom to top we obtain a permutation whose descent set is as prescribed.
Code
def composition_to_ribbon(c):
    inner = []
    outer = []
    indent = 0
    for p in reversed(c):
        if indent > 0:
            inner.append(indent)
        outer.append(p+indent)
        indent += p-1
    return SkewPartition([outer[::-1], inner[::-1]])

def statistic(c):
    return StandardSkewTableaux(composition_to_ribbon(c)).cardinality()

# alternative implementation

def descents_composition(elt):
    if len(elt) == 0:
        return Composition([])
    d = [-1] + elt.descents() + [len(elt)-1]
    return Composition([ d[i+1]-d[i] for i in range(len(d)-1)])

@cached_function
def preimages(level):
    result = dict()
    for el in Permutations(level):
        image = descents_composition(el)
        result[image] = result.get(image, 0) + 1
    return result

def statistic(x):
    return preimages(x.size()).get(x, 0)

Created
Sep 11, 2015 at 21:12 by Martin Rubey
Updated
Sep 16, 2019 at 09:44 by Martin Rubey