Identifier
Identifier
Values
[1] => 1
[1,2] => 1
[2,1] => 1
[1,2,3] => 1
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 2
[3,1,2] => 2
[3,2,1] => 5
[1,2,3,4] => 1
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 5
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 6
[2,4,1,3] => 6
[2,4,3,1] => 18
[3,1,2,4] => 2
[3,1,4,2] => 6
[3,2,1,4] => 5
[3,2,4,1] => 18
[3,4,1,2] => 22
[3,4,2,1] => 62
[4,1,2,3] => 6
[4,1,3,2] => 18
[4,2,1,3] => 18
[4,2,3,1] => 73
[4,3,1,2] => 62
[4,3,2,1] => 210
[1,2,3,4,5] => 1
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 5
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 6
[1,3,5,2,4] => 6
[1,3,5,4,2] => 18
[1,4,2,3,5] => 2
[1,4,2,5,3] => 6
[1,4,3,2,5] => 5
[1,4,3,5,2] => 18
[1,4,5,2,3] => 22
[1,4,5,3,2] => 62
[1,5,2,3,4] => 6
[1,5,2,4,3] => 18
[1,5,3,2,4] => 18
[1,5,3,4,2] => 73
[1,5,4,2,3] => 62
[1,5,4,3,2] => 210
[2,1,3,4,5] => 1
[2,1,3,5,4] => 2
[2,1,4,3,5] => 2
[2,1,4,5,3] => 6
[2,1,5,3,4] => 6
[2,1,5,4,3] => 18
[2,3,1,4,5] => 2
[2,3,1,5,4] => 6
[2,3,4,1,5] => 6
[2,3,4,5,1] => 24
[2,3,5,1,4] => 24
[2,3,5,4,1] => 86
[2,4,1,3,5] => 6
[2,4,1,5,3] => 24
[2,4,3,1,5] => 18
[2,4,3,5,1] => 86
[2,4,5,1,3] => 106
[2,4,5,3,1] => 352
[2,5,1,3,4] => 24
[2,5,1,4,3] => 86
[2,5,3,1,4] => 86
[2,5,3,4,1] => 418
[2,5,4,1,3] => 352
[2,5,4,3,1] => 1382
[3,1,2,4,5] => 2
[3,1,2,5,4] => 6
[3,1,4,2,5] => 6
[3,1,4,5,2] => 24
[3,1,5,2,4] => 24
[3,1,5,4,2] => 86
[3,2,1,4,5] => 5
[3,2,1,5,4] => 18
[3,2,4,1,5] => 18
[3,2,4,5,1] => 86
[3,2,5,1,4] => 86
[3,2,5,4,1] => 354
[3,4,1,2,5] => 22
[3,4,1,5,2] => 106
[3,4,2,1,5] => 62
[3,4,2,5,1] => 352
[3,4,5,1,2] => 508
[3,4,5,2,1] => 1634
[3,5,1,2,4] => 106
[3,5,1,4,2] => 462
[3,5,2,1,4] => 352
[3,5,2,4,1] => 1958
[3,5,4,1,2] => 1926
[3,5,4,2,1] => 7160
[4,1,2,3,5] => 6
[4,1,2,5,3] => 24
[4,1,3,2,5] => 18
[4,1,3,5,2] => 86
[4,1,5,2,3] => 106
[4,1,5,3,2] => 352
[4,2,1,3,5] => 18
[4,2,1,5,3] => 86
[4,2,3,1,5] => 73
[4,2,3,5,1] => 418
[4,2,5,1,3] => 462
[4,2,5,3,1] => 1958
[4,3,1,2,5] => 62
[4,3,1,5,2] => 352
[4,3,2,1,5] => 210
[4,3,2,5,1] => 1382
[4,3,5,1,2] => 1926
[4,3,5,2,1] => 7160
[4,5,1,2,3] => 508
[4,5,1,3,2] => 1926
[4,5,2,1,3] => 1926
[4,5,2,3,1] => 10544
[4,5,3,1,2] => 8022
[4,5,3,2,1] => 34266
[5,1,2,3,4] => 24
[5,1,2,4,3] => 86
[5,1,3,2,4] => 86
[5,1,3,4,2] => 418
[5,1,4,2,3] => 352
[5,1,4,3,2] => 1382
[5,2,1,3,4] => 86
[5,2,1,4,3] => 354
[5,2,3,1,4] => 418
[5,2,3,4,1] => 2381
[5,2,4,1,3] => 1958
[5,2,4,3,1] => 9234
[5,3,1,2,4] => 352
[5,3,1,4,2] => 1958
[5,3,2,1,4] => 1382
[5,3,2,4,1] => 9234
[5,3,4,1,2] => 10544
[5,3,4,2,1] => 44062
[5,4,1,2,3] => 1634
[5,4,1,3,2] => 7160
[5,4,2,1,3] => 7160
[5,4,2,3,1] => 44062
[5,4,3,1,2] => 34266
[5,4,3,2,1] => 162482
[1,2,3,4,5,6] => 1
[1,2,3,4,6,5] => 1
[1,2,3,5,4,6] => 1
[1,2,3,5,6,4] => 2
[1,2,3,6,4,5] => 2
[1,2,3,6,5,4] => 5
[1,2,4,3,5,6] => 1
[1,2,4,3,6,5] => 2
[1,2,4,5,3,6] => 2
[1,2,4,5,6,3] => 6
[1,2,4,6,3,5] => 6
[1,2,4,6,5,3] => 18
[1,2,5,3,4,6] => 2
[1,2,5,3,6,4] => 6
[1,2,5,4,3,6] => 5
[1,2,5,4,6,3] => 18
[1,2,5,6,3,4] => 22
[1,2,5,6,4,3] => 62
[1,2,6,3,4,5] => 6
[1,2,6,3,5,4] => 18
[1,2,6,4,3,5] => 18
[1,2,6,4,5,3] => 73
[1,2,6,5,3,4] => 62
[1,2,6,5,4,3] => 210
[1,3,2,4,5,6] => 1
[1,3,2,4,6,5] => 2
[1,3,2,5,4,6] => 2
[1,3,2,5,6,4] => 6
[1,3,2,6,4,5] => 6
[1,3,2,6,5,4] => 18
[1,3,4,2,5,6] => 2
[1,3,4,2,6,5] => 6
[1,3,4,5,2,6] => 6
[1,3,4,5,6,2] => 24
[1,3,4,6,2,5] => 24
[1,3,4,6,5,2] => 86
[1,3,5,2,4,6] => 6
[1,3,5,2,6,4] => 24
[1,3,5,4,2,6] => 18
[1,3,5,4,6,2] => 86
[1,3,5,6,2,4] => 106
[1,3,5,6,4,2] => 352
[1,3,6,2,4,5] => 24
[1,3,6,2,5,4] => 86
[1,3,6,4,2,5] => 86
[1,3,6,4,5,2] => 418
[1,3,6,5,2,4] => 352
[1,3,6,5,4,2] => 1382
[1,4,2,3,5,6] => 2
[1,4,2,3,6,5] => 6
[1,4,2,5,3,6] => 6
[1,4,2,5,6,3] => 24
[1,4,2,6,3,5] => 24
[1,4,2,6,5,3] => 86
[1,4,3,2,5,6] => 5
[1,4,3,2,6,5] => 18
[1,4,3,5,2,6] => 18
[1,4,3,5,6,2] => 86
[1,4,3,6,2,5] => 86
[1,4,3,6,5,2] => 354
[1,4,5,2,3,6] => 22
[1,4,5,2,6,3] => 106
[1,4,5,3,2,6] => 62
[1,4,5,3,6,2] => 352
[1,4,5,6,2,3] => 508
[1,4,5,6,3,2] => 1634
[1,4,6,2,3,5] => 106
[1,4,6,2,5,3] => 462
[1,4,6,3,2,5] => 352
[1,4,6,3,5,2] => 1958
[1,4,6,5,2,3] => 1926
[1,4,6,5,3,2] => 7160
[1,5,2,3,4,6] => 6
[1,5,2,3,6,4] => 24
[1,5,2,4,3,6] => 18
[1,5,2,4,6,3] => 86
[1,5,2,6,3,4] => 106
[1,5,2,6,4,3] => 352
[1,5,3,2,4,6] => 18
[1,5,3,2,6,4] => 86
[1,5,3,4,2,6] => 73
[1,5,3,4,6,2] => 418
[1,5,3,6,2,4] => 462
[1,5,3,6,4,2] => 1958
[1,5,4,2,3,6] => 62
[1,5,4,2,6,3] => 352
[1,5,4,3,2,6] => 210
[1,5,4,3,6,2] => 1382
[1,5,4,6,2,3] => 1926
[1,5,4,6,3,2] => 7160
[1,5,6,2,3,4] => 508
[1,5,6,2,4,3] => 1926
[1,5,6,3,2,4] => 1926
[1,5,6,3,4,2] => 10544
[1,5,6,4,2,3] => 8022
[1,5,6,4,3,2] => 34266
[1,6,2,3,4,5] => 24
[1,6,2,3,5,4] => 86
[1,6,2,4,3,5] => 86
[1,6,2,4,5,3] => 418
[1,6,2,5,3,4] => 352
[1,6,2,5,4,3] => 1382
[1,6,3,2,4,5] => 86
[1,6,3,2,5,4] => 354
[1,6,3,4,2,5] => 418
[1,6,3,4,5,2] => 2381
[1,6,3,5,2,4] => 1958
[1,6,3,5,4,2] => 9234
[1,6,4,2,3,5] => 352
[1,6,4,2,5,3] => 1958
[1,6,4,3,2,5] => 1382
[1,6,4,3,5,2] => 9234
[1,6,4,5,2,3] => 10544
[1,6,4,5,3,2] => 44062
[1,6,5,2,3,4] => 1634
[1,6,5,2,4,3] => 7160
[1,6,5,3,2,4] => 7160
[1,6,5,3,4,2] => 44062
[1,6,5,4,2,3] => 34266
[1,6,5,4,3,2] => 162482
[2,1,3,4,5,6] => 1
[2,1,3,4,6,5] => 2
[2,1,3,5,4,6] => 2
[2,1,3,5,6,4] => 6
[2,1,3,6,4,5] => 6
[2,1,3,6,5,4] => 18
[2,1,4,3,5,6] => 2
[2,1,4,3,6,5] => 6
[2,1,4,5,3,6] => 6
[2,1,4,5,6,3] => 24
[2,1,4,6,3,5] => 24
[2,1,4,6,5,3] => 86
[2,1,5,3,4,6] => 6
[2,1,5,3,6,4] => 24
[2,1,5,4,3,6] => 18
[2,1,5,4,6,3] => 86
[2,1,5,6,3,4] => 106
[2,1,5,6,4,3] => 352
[2,1,6,3,4,5] => 24
[2,1,6,3,5,4] => 86
[2,1,6,4,3,5] => 86
[2,1,6,4,5,3] => 418
[2,1,6,5,3,4] => 352
[2,1,6,5,4,3] => 1382
[2,3,1,4,5,6] => 2
[2,3,1,4,6,5] => 6
[2,3,1,5,4,6] => 6
[2,3,1,5,6,4] => 24
[2,3,1,6,4,5] => 24
[2,3,1,6,5,4] => 86
[2,3,4,1,5,6] => 6
[2,3,4,1,6,5] => 24
[2,3,4,5,1,6] => 24
[2,3,4,5,6,1] => 120
[2,3,4,6,1,5] => 120
[2,3,4,6,5,1] => 504
[2,3,5,1,4,6] => 24
[2,3,5,1,6,4] => 120
[2,3,5,4,1,6] => 86
[2,3,5,4,6,1] => 504
[2,3,5,6,1,4] => 624
[2,3,5,6,4,1] => 2384
[2,3,6,1,4,5] => 120
[2,3,6,1,5,4] => 504
[2,3,6,4,1,5] => 504
[2,3,6,4,5,1] => 2854
[2,3,6,5,1,4] => 2384
[2,3,6,5,4,1] => 10632
[2,4,1,3,5,6] => 6
[2,4,1,3,6,5] => 24
[2,4,1,5,3,6] => 24
[2,4,1,5,6,3] => 120
[2,4,1,6,3,5] => 120
[2,4,1,6,5,3] => 504
[2,4,3,1,5,6] => 18
[2,4,3,1,6,5] => 86
[2,4,3,5,1,6] => 86
[2,4,3,5,6,1] => 504
[2,4,3,6,1,5] => 504
[2,4,3,6,5,1] => 2406
[2,4,5,1,3,6] => 106
[2,4,5,1,6,3] => 624
[2,4,5,3,1,6] => 352
[2,4,5,3,6,1] => 2384
[2,4,5,6,1,3] => 3468
[2,4,5,6,3,1] => 12608
[2,4,6,1,3,5] => 624
[2,4,6,1,5,3] => 3154
[2,4,6,3,1,5] => 2384
[2,4,6,3,5,1] => 15232
[2,4,6,5,1,3] => 14904
[2,4,6,5,3,1] => 62040
[2,5,1,3,4,6] => 24
[2,5,1,3,6,4] => 120
[2,5,1,4,3,6] => 86
[2,5,1,4,6,3] => 504
[2,5,1,6,3,4] => 624
[2,5,1,6,4,3] => 2384
[2,5,3,1,4,6] => 86
[2,5,3,1,6,4] => 504
[2,5,3,4,1,6] => 418
[2,5,3,4,6,1] => 2854
[2,5,3,6,1,4] => 3154
[2,5,3,6,4,1] => 15232
[2,5,4,1,3,6] => 352
[2,5,4,1,6,3] => 2384
[2,5,4,3,1,6] => 1382
[2,5,4,3,6,1] => 10632
[2,5,4,6,1,3] => 14904
[2,5,4,6,3,1] => 62040
[2,5,6,1,3,4] => 3468
[2,5,6,1,4,3] => 14904
[2,5,6,3,1,4] => 14904
[2,5,6,3,4,1] => 92128
[2,5,6,4,1,3] => 69482
[2,5,6,4,3,1] => 329304
[2,6,1,3,4,5] => 120
[2,6,1,3,5,4] => 504
[2,6,1,4,3,5] => 504
[2,6,1,4,5,3] => 2854
[2,6,1,5,3,4] => 2384
[2,6,1,5,4,3] => 10632
[2,6,3,1,4,5] => 504
[2,6,3,1,5,4] => 2406
[2,6,3,4,1,5] => 2854
[2,6,3,4,5,1] => 18610
[2,6,3,5,1,4] => 15232
[2,6,3,5,4,1] => 80582
[2,6,4,1,3,5] => 2384
[2,6,4,1,5,3] => 15232
[2,6,4,3,1,5] => 10632
[2,6,4,3,5,1] => 80582
[2,6,4,5,1,3] => 92128
[2,6,4,5,3,1] => 426328
[2,6,5,1,3,4] => 12608
[2,6,5,1,4,3] => 62040
[2,6,5,3,1,4] => 62040
[2,6,5,3,4,1] => 426328
[2,6,5,4,1,3] => 329304
[2,6,5,4,3,1] => 1718598
[3,1,2,4,5,6] => 2
[3,1,2,4,6,5] => 6
[3,1,2,5,4,6] => 6
[3,1,2,5,6,4] => 24
[3,1,2,6,4,5] => 24
[3,1,2,6,5,4] => 86
[3,1,4,2,5,6] => 6
[3,1,4,2,6,5] => 24
[3,1,4,5,2,6] => 24
[3,1,4,5,6,2] => 120
[3,1,4,6,2,5] => 120
[3,1,4,6,5,2] => 504
[3,1,5,2,4,6] => 24
[3,1,5,2,6,4] => 120
[3,1,5,4,2,6] => 86
[3,1,5,4,6,2] => 504
[3,1,5,6,2,4] => 624
[3,1,5,6,4,2] => 2384
[3,1,6,2,4,5] => 120
[3,1,6,2,5,4] => 504
[3,1,6,4,2,5] => 504
[3,1,6,4,5,2] => 2854
[3,1,6,5,2,4] => 2384
[3,1,6,5,4,2] => 10632
[3,2,1,4,5,6] => 5
[3,2,1,4,6,5] => 18
[3,2,1,5,4,6] => 18
[3,2,1,5,6,4] => 86
[3,2,1,6,4,5] => 86
[3,2,1,6,5,4] => 354
[3,2,4,1,5,6] => 18
[3,2,4,1,6,5] => 86
[3,2,4,5,1,6] => 86
[3,2,4,5,6,1] => 504
[3,2,4,6,1,5] => 504
[3,2,4,6,5,1] => 2406
[3,2,5,1,4,6] => 86
[3,2,5,1,6,4] => 504
[3,2,5,4,1,6] => 354
[3,2,5,4,6,1] => 2406
[3,2,5,6,1,4] => 2986
[3,2,5,6,4,1] => 12800
[3,2,6,1,4,5] => 504
[3,2,6,1,5,4] => 2406
[3,2,6,4,1,5] => 2406
[3,2,6,4,5,1] => 15410
[3,2,6,5,1,4] => 12800
[3,2,6,5,4,1] => 63510
[3,4,1,2,5,6] => 22
[3,4,1,2,6,5] => 106
[3,4,1,5,2,6] => 106
[3,4,1,5,6,2] => 624
[3,4,1,6,2,5] => 624
[3,4,1,6,5,2] => 2986
[3,4,2,1,5,6] => 62
[3,4,2,1,6,5] => 352
[3,4,2,5,1,6] => 352
[3,4,2,5,6,1] => 2384
[3,4,2,6,1,5] => 2384
[3,4,2,6,5,1] => 12800
[3,4,5,1,2,6] => 508
[3,4,5,1,6,2] => 3468
[3,4,5,2,1,6] => 1634
[3,4,5,2,6,1] => 12608
[3,4,5,6,1,2] => 20944
[3,4,5,6,2,1] => 74024
[3,4,6,1,2,5] => 3468
[3,4,6,1,5,2] => 20356
[3,4,6,2,1,5] => 12608
[3,4,6,2,5,1] => 90002
[3,4,6,5,1,2] => 100380
[3,4,6,5,2,1] => 400224
[3,5,1,2,4,6] => 106
[3,5,1,2,6,4] => 624
[3,5,1,4,2,6] => 462
[3,5,1,4,6,2] => 3154
[3,5,1,6,2,4] => 3774
[3,5,1,6,4,2] => 16832
[3,5,2,1,4,6] => 352
[3,5,2,1,6,4] => 2384
[3,5,2,4,1,6] => 1958
[3,5,2,4,6,1] => 15232
[3,5,2,6,1,4] => 16832
[3,5,2,6,4,1] => 90456
[3,5,4,1,2,6] => 1926
[3,5,4,1,6,2] => 14904
[3,5,4,2,1,6] => 7160
[3,5,4,2,6,1] => 62040
[3,5,4,6,1,2] => 99744
[3,5,4,6,2,1] => 398456
[3,5,6,1,2,4] => 21736
[3,5,6,1,4,2] => 110764
[3,5,6,2,1,4] => 87676
[3,5,6,2,4,1] => 598728
[3,5,6,4,1,2] => 512684
[3,5,6,4,2,1] => 2300762
[3,6,1,2,4,5] => 624
[3,6,1,2,5,4] => 2986
[3,6,1,4,2,5] => 3154
[3,6,1,4,5,2] => 20374
[3,6,1,5,2,4] => 16832
[3,6,1,5,4,2] => 85802
[3,6,2,1,4,5] => 2384
[3,6,2,1,5,4] => 12800
[3,6,2,4,1,5] => 15232
[3,6,2,4,5,1] => 110878
[3,6,2,5,1,4] => 90456
[3,6,2,5,4,1] => 527840
[3,6,4,1,2,5] => 14904
[3,6,4,1,5,2] => 108326
[3,6,4,2,1,5] => 62040
[3,6,4,2,5,1] => 521224
[3,6,4,5,1,2] => 683144
[3,6,4,5,2,1] => 3000024
[3,6,5,1,2,4] => 87676
[3,6,5,1,4,2] => 506992
[3,6,5,2,1,4] => 399248
[3,6,5,2,4,1] => 3013576
[3,6,5,4,1,2] => 2651262
[3,6,5,4,2,1] => 12996712
[4,1,2,3,5,6] => 6
[4,1,2,3,6,5] => 24
[4,1,2,5,3,6] => 24
[4,1,2,5,6,3] => 120
[4,1,2,6,3,5] => 120
[4,1,2,6,5,3] => 504
[4,1,3,2,5,6] => 18
[4,1,3,2,6,5] => 86
[4,1,3,5,2,6] => 86
[4,1,3,5,6,2] => 504
[4,1,3,6,2,5] => 504
[4,1,3,6,5,2] => 2406
[4,1,5,2,3,6] => 106
[4,1,5,2,6,3] => 624
[4,1,5,3,2,6] => 352
[4,1,5,3,6,2] => 2384
[4,1,5,6,2,3] => 3468
[4,1,5,6,3,2] => 12608
[4,1,6,2,3,5] => 624
[4,1,6,2,5,3] => 3154
[4,1,6,3,2,5] => 2384
[4,1,6,3,5,2] => 15232
[4,1,6,5,2,3] => 14904
[4,1,6,5,3,2] => 62040
[4,2,1,3,5,6] => 18
[4,2,1,3,6,5] => 86
[4,2,1,5,3,6] => 86
[4,2,1,5,6,3] => 504
[4,2,1,6,3,5] => 504
[4,2,1,6,5,3] => 2406
[4,2,3,1,5,6] => 73
[4,2,3,1,6,5] => 418
[4,2,3,5,1,6] => 418
[4,2,3,5,6,1] => 2854
[4,2,3,6,1,5] => 2854
[4,2,3,6,5,1] => 15410
[4,2,5,1,3,6] => 462
[4,2,5,1,6,3] => 3154
[4,2,5,3,1,6] => 1958
[4,2,5,3,6,1] => 15232
[4,2,5,6,1,3] => 20356
[4,2,5,6,3,1] => 90002
[4,2,6,1,3,5] => 3154
[4,2,6,1,5,3] => 17574
[4,2,6,3,1,5] => 15232
[4,2,6,3,5,1] => 109358
[4,2,6,5,1,3] => 96902
[4,2,6,5,3,1] => 490008
[4,3,1,2,5,6] => 62
[4,3,1,2,6,5] => 352
[4,3,1,5,2,6] => 352
[4,3,1,5,6,2] => 2384
[4,3,1,6,2,5] => 2384
[4,3,1,6,5,2] => 12800
[4,3,2,1,5,6] => 210
[4,3,2,1,6,5] => 1382
[4,3,2,5,1,6] => 1382
[4,3,2,5,6,1] => 10632
[4,3,2,6,1,5] => 10632
[4,3,2,6,5,1] => 63510
[4,3,5,1,2,6] => 1926
[4,3,5,1,6,2] => 14904
[4,3,5,2,1,6] => 7160
[4,3,5,2,6,1] => 62040
[4,3,5,6,1,2] => 100380
[4,3,5,6,2,1] => 400224
[4,3,6,1,2,5] => 14904
[4,3,6,1,5,2] => 96902
[4,3,6,2,1,5] => 62040
[4,3,6,2,5,1] => 490008
[4,3,6,5,1,2] => 526176
[4,3,6,5,2,1] => 2358416
[4,5,1,2,3,6] => 508
[4,5,1,2,6,3] => 3468
[4,5,1,3,2,6] => 1926
[4,5,1,3,6,2] => 14904
[4,5,1,6,2,3] => 21736
[4,5,1,6,3,2] => 87676
[4,5,2,1,3,6] => 1926
[4,5,2,1,6,3] => 14904
[4,5,2,3,1,6] => 10544
[4,5,2,3,6,1] => 92128
[4,5,2,6,1,3] => 110764
[4,5,2,6,3,1] => 598728
[4,5,3,1,2,6] => 8022
[4,5,3,1,6,2] => 69482
[4,5,3,2,1,6] => 34266
[4,5,3,2,6,1] => 329304
[4,5,3,6,1,2] => 512684
[4,5,3,6,2,1] => 2300762
[4,5,6,1,2,3] => 136706
[4,5,6,1,3,2] => 606384
[4,5,6,2,1,3] => 606384
[4,5,6,2,3,1] => 4050488
[4,5,6,3,1,2] => 2845672
[4,5,6,3,2,1] => 14292656
[4,6,1,2,3,5] => 3468
[4,6,1,2,5,3] => 20356
[4,6,1,3,2,5] => 14904
[4,6,1,3,5,2] => 108326
[4,6,1,5,2,3] => 110764
[4,6,1,5,3,2] => 506992
[4,6,2,1,3,5] => 14904
[4,6,2,1,5,3] => 96902
[4,6,2,3,1,5] => 92128
[4,6,2,3,5,1] => 753904
[4,6,2,5,1,3] => 692928
[4,6,2,5,3,1] => 4022640
[4,6,3,1,2,5] => 69482
[4,6,3,1,5,2] => 556686
[4,6,3,2,1,5] => 329304
[4,6,3,2,5,1] => 3035178
[4,6,3,5,1,2] => 3833302
[4,6,3,5,2,1] => 18816048
[4,6,5,1,2,3] => 606384
[4,6,5,1,3,2] => 3001406
[4,6,5,2,1,3] => 3019208
[4,6,5,2,3,1] => 22018344
[4,6,5,3,1,2] => 15762128
[4,6,5,3,2,1] => 86110376
[5,1,2,3,4,6] => 24
[5,1,2,3,6,4] => 120
[5,1,2,4,3,6] => 86
[5,1,2,4,6,3] => 504
[5,1,2,6,3,4] => 624
[5,1,2,6,4,3] => 2384
[5,1,3,2,4,6] => 86
[5,1,3,2,6,4] => 504
[5,1,3,4,2,6] => 418
[5,1,3,4,6,2] => 2854
[5,1,3,6,2,4] => 3154
[5,1,3,6,4,2] => 15232
[5,1,4,2,3,6] => 352
[5,1,4,2,6,3] => 2384
[5,1,4,3,2,6] => 1382
[5,1,4,3,6,2] => 10632
[5,1,4,6,2,3] => 14904
[5,1,4,6,3,2] => 62040
[5,1,6,2,3,4] => 3468
[5,1,6,2,4,3] => 14904
[5,1,6,3,2,4] => 14904
[5,1,6,3,4,2] => 92128
[5,1,6,4,2,3] => 69482
[5,1,6,4,3,2] => 329304
[5,2,1,3,4,6] => 86
[5,2,1,3,6,4] => 504
[5,2,1,4,3,6] => 354
[5,2,1,4,6,3] => 2406
[5,2,1,6,3,4] => 2986
[5,2,1,6,4,3] => 12800
[5,2,3,1,4,6] => 418
[5,2,3,1,6,4] => 2854
[5,2,3,4,1,6] => 2381
[5,2,3,4,6,1] => 18610
[5,2,3,6,1,4] => 20374
[5,2,3,6,4,1] => 110878
[5,2,4,1,3,6] => 1958
[5,2,4,1,6,3] => 15232
[5,2,4,3,1,6] => 9234
[5,2,4,3,6,1] => 80582
[5,2,4,6,1,3] => 108326
[5,2,4,6,3,1] => 521224
[5,2,6,1,3,4] => 20356
[5,2,6,1,4,3] => 96902
[5,2,6,3,1,4] => 108326
[5,2,6,3,4,1] => 753904
[5,2,6,4,1,3] => 556686
[5,2,6,4,3,1] => 3035178
[5,3,1,2,4,6] => 352
[5,3,1,2,6,4] => 2384
[5,3,1,4,2,6] => 1958
[5,3,1,4,6,2] => 15232
[5,3,1,6,2,4] => 16832
[5,3,1,6,4,2] => 90456
[5,3,2,1,4,6] => 1382
[5,3,2,1,6,4] => 10632
[5,3,2,4,1,6] => 9234
[5,3,2,4,6,1] => 80582
[5,3,2,6,1,4] => 85802
[5,3,2,6,4,1] => 527840
[5,3,4,1,2,6] => 10544
[5,3,4,1,6,2] => 92128
[5,3,4,2,1,6] => 44062
[5,3,4,2,6,1] => 426328
[5,3,4,6,1,2] => 683144
[5,3,4,6,2,1] => 3000024
[5,3,6,1,2,4] => 110764
[5,3,6,1,4,2] => 692928
[5,3,6,2,1,4] => 506992
[5,3,6,2,4,1] => 4022640
[5,3,6,4,1,2] => 3833302
[5,3,6,4,2,1] => 18816048
[5,4,1,2,3,6] => 1634
[5,4,1,2,6,3] => 12608
[5,4,1,3,2,6] => 7160
[5,4,1,3,6,2] => 62040
[5,4,1,6,2,3] => 87676
[5,4,1,6,3,2] => 399248
[5,4,2,1,3,6] => 7160
[5,4,2,1,6,3] => 62040
[5,4,2,3,1,6] => 44062
[5,4,2,3,6,1] => 426328
[5,4,2,6,1,3] => 506992
[5,4,2,6,3,1] => 3013576
[5,4,3,1,2,6] => 34266
[5,4,3,1,6,2] => 329304
[5,4,3,2,1,6] => 162482
[5,4,3,2,6,1] => 1718598
[5,4,3,6,1,2] => 2651262
[5,4,3,6,2,1] => 12996712
[5,4,6,1,2,3] => 606384
[5,4,6,1,3,2] => 3019208
[5,4,6,2,1,3] => 3001406
[5,4,6,2,3,1] => 22018344
[5,4,6,3,1,2] => 15762128
[5,4,6,3,2,1] => 86110376
[5,6,1,2,3,4] => 20944
[5,6,1,2,4,3] => 100380
[5,6,1,3,2,4] => 99744
[5,6,1,3,4,2] => 683144
[5,6,1,4,2,3] => 512684
[5,6,1,4,3,2] => 2651262
[5,6,2,1,3,4] => 100380
[5,6,2,1,4,3] => 526176
[5,6,2,3,1,4] => 683144
[5,6,2,3,4,1] => 5308656
[5,6,2,4,1,3] => 3833302
[5,6,2,4,3,1] => 23875224
[5,6,3,1,2,4] => 512684
[5,6,3,1,4,2] => 3833302
[5,6,3,2,1,4] => 2651262
[5,6,3,2,4,1] => 23875224
[5,6,3,4,1,2] => 25948726
[5,6,3,4,2,1] => 138388118
[5,6,4,1,2,3] => 2845672
[5,6,4,1,3,2] => 15762128
[5,6,4,2,1,3] => 15762128
[5,6,4,2,3,1] => 125228144
[5,6,4,3,1,2] => 91769546
[5,6,4,3,2,1] => 542024608
[6,1,2,3,4,5] => 120
[6,1,2,3,5,4] => 504
[6,1,2,4,3,5] => 504
[6,1,2,4,5,3] => 2854
[6,1,2,5,3,4] => 2384
[6,1,2,5,4,3] => 10632
[6,1,3,2,4,5] => 504
[6,1,3,2,5,4] => 2406
[6,1,3,4,2,5] => 2854
[6,1,3,4,5,2] => 18610
[6,1,3,5,2,4] => 15232
[6,1,3,5,4,2] => 80582
[6,1,4,2,3,5] => 2384
[6,1,4,2,5,3] => 15232
[6,1,4,3,2,5] => 10632
[6,1,4,3,5,2] => 80582
[6,1,4,5,2,3] => 92128
[6,1,4,5,3,2] => 426328
[6,1,5,2,3,4] => 12608
[6,1,5,2,4,3] => 62040
[6,1,5,3,2,4] => 62040
[6,1,5,3,4,2] => 426328
[6,1,5,4,2,3] => 329304
[6,1,5,4,3,2] => 1718598
[6,2,1,3,4,5] => 504
[6,2,1,3,5,4] => 2406
[6,2,1,4,3,5] => 2406
[6,2,1,4,5,3] => 15410
[6,2,1,5,3,4] => 12800
[6,2,1,5,4,3] => 63510
[6,2,3,1,4,5] => 2854
[6,2,3,1,5,4] => 15410
[6,2,3,4,1,5] => 18610
[6,2,3,4,5,1] => 136081
[6,2,3,5,1,4] => 110878
[6,2,3,5,4,1] => 651730
[6,2,4,1,3,5] => 15232
[6,2,4,1,5,3] => 109358
[6,2,4,3,1,5] => 80582
[6,2,4,3,5,1] => 681938
[6,2,4,5,1,3] => 753904
[6,2,4,5,3,1] => 3979998
[6,2,5,1,3,4] => 90002
[6,2,5,1,4,3] => 490008
[6,2,5,3,1,4] => 521224
[6,2,5,3,4,1] => 3979998
[6,2,5,4,1,3] => 3035178
[6,2,5,4,3,1] => 17788450
[6,3,1,2,4,5] => 2384
[6,3,1,2,5,4] => 12800
[6,3,1,4,2,5] => 15232
[6,3,1,4,5,2] => 110878
[6,3,1,5,2,4] => 90456
[6,3,1,5,4,2] => 527840
[6,3,2,1,4,5] => 10632
[6,3,2,1,5,4] => 63510
[6,3,2,4,1,5] => 80582
[6,3,2,4,5,1] => 651730
[6,3,2,5,1,4] => 527840
[6,3,2,5,4,1] => 3457350
[6,3,4,1,2,5] => 92128
[6,3,4,1,5,2] => 753904
[6,3,4,2,1,5] => 426328
[6,3,4,2,5,1] => 3979998
[6,3,4,5,1,2] => 5308656
[6,3,4,5,2,1] => 25359304
[6,3,5,1,2,4] => 598728
[6,3,5,1,4,2] => 4022640
[6,3,5,2,1,4] => 3013576
[6,3,5,2,4,1] => 25618632
[6,3,5,4,1,2] => 23875224
[6,3,5,4,2,1] => 126450110
[6,4,1,2,3,5] => 12608
[6,4,1,2,5,3] => 90002
[6,4,1,3,2,5] => 62040
[6,4,1,3,5,2] => 521224
[6,4,1,5,2,3] => 598728
[6,4,1,5,3,2] => 3013576
[6,4,2,1,3,5] => 62040
[6,4,2,1,5,3] => 490008
[6,4,2,3,1,5] => 426328
[6,4,2,3,5,1] => 3979998
[6,4,2,5,1,3] => 4022640
[6,4,2,5,3,1] => 25618632
[6,4,3,1,2,5] => 329304
[6,4,3,1,5,2] => 3035178
[6,4,3,2,1,5] => 1718598
[6,4,3,2,5,1] => 17788450
[6,4,3,5,1,2] => 23875224
[6,4,3,5,2,1] => 126450110
[6,4,5,1,2,3] => 4050488
[6,4,5,1,3,2] => 22018344
[6,4,5,2,1,3] => 22018344
[6,4,5,2,3,1] => 174937994
[6,4,5,3,1,2] => 125228144
[6,4,5,3,2,1] => 737498144
[6,5,1,2,3,4] => 74024
[6,5,1,2,4,3] => 400224
[6,5,1,3,2,4] => 398456
[6,5,1,3,4,2] => 3000024
[6,5,1,4,2,3] => 2300762
[6,5,1,4,3,2] => 12996712
[6,5,2,1,3,4] => 400224
[6,5,2,1,4,3] => 2358416
[6,5,2,3,1,4] => 3000024
[6,5,2,3,4,1] => 25359304
[6,5,2,4,1,3] => 18816048
[6,5,2,4,3,1] => 126450110
[6,5,3,1,2,4] => 2300762
[6,5,3,1,4,2] => 18816048
[6,5,3,2,1,4] => 12996712
[6,5,3,2,4,1] => 126450110
[6,5,3,4,1,2] => 138388118
[6,5,3,4,2,1] => 791921586
[6,5,4,1,2,3] => 14292656
[6,5,4,1,3,2] => 86110376
[6,5,4,2,1,3] => 86110376
[6,5,4,2,3,1] => 737498144
[6,5,4,3,1,2] => 542024608
click to show generating function       
Description
The number of strong Bruhat factorizations of a permutation.
This is, the number of factorizations $\pi = t_1 \cdots t_\ell$ for transpositions $\{ t_i \mid 1 \leq i \leq \ell\}$ such that $t_1 \cdots t_i$ has more inversions than $t_1 \cdots t_{i-1}$ for all $1 \leq i \leq \ell$.
Code
@cached_function
def Transpositions(n):
    return Permutations(n).conjugacy_class([2]+[1]*(n-2))

@cached_function
def strong_Bruhat_factorizations(pi):
    inv = pi.number_of_inversions()
    if inv == 0:
        return 1
    else:
        facs = 0
        for t in Transpositions(len(pi)):
            tau = pi*t
            if tau.number_of_inversions() < inv:
                facs += strong_Bruhat_factorizations(tau)
        return facs

def statistic(pi):
    return strong_Bruhat_factorizations(pi)
Created
Aug 28, 2017 at 16:39 by Christian Stump
Updated
Aug 28, 2017 at 16:39 by Christian Stump