Identifier
Identifier
Values
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 1
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 1
[3,4,1,2] => 1
[3,4,2,1] => 2
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 4
[4,3,1,2] => 2
[4,3,2,1] => 10
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 1
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 0
[1,4,2,5,3] => 1
[1,4,3,2,5] => 0
[1,4,3,5,2] => 1
[1,4,5,2,3] => 1
[1,4,5,3,2] => 2
[1,5,2,3,4] => 0
[1,5,2,4,3] => 1
[1,5,3,2,4] => 1
[1,5,3,4,2] => 4
[1,5,4,2,3] => 2
[1,5,4,3,2] => 10
[2,1,3,4,5] => 0
[2,1,3,5,4] => 1
[2,1,4,3,5] => 1
[2,1,4,5,3] => 2
[2,1,5,3,4] => 2
[2,1,5,4,3] => 6
[2,3,1,4,5] => 0
[2,3,1,5,4] => 2
[2,3,4,1,5] => 0
[2,3,4,5,1] => 0
[2,3,5,1,4] => 2
[2,3,5,4,1] => 2
[2,4,1,3,5] => 1
[2,4,1,5,3] => 5
[2,4,3,1,5] => 1
[2,4,3,5,1] => 2
[2,4,5,1,3] => 5
[2,4,5,3,1] => 7
[2,5,1,3,4] => 2
[2,5,1,4,3] => 11
[2,5,3,1,4] => 4
[2,5,3,4,1] => 9
[2,5,4,1,3] => 16
[2,5,4,3,1] => 37
[3,1,2,4,5] => 0
[3,1,2,5,4] => 2
[3,1,4,2,5] => 1
[3,1,4,5,2] => 2
[3,1,5,2,4] => 5
[3,1,5,4,2] => 11
[3,2,1,4,5] => 0
[3,2,1,5,4] => 6
[3,2,4,1,5] => 1
[3,2,4,5,1] => 2
[3,2,5,1,4] => 11
[3,2,5,4,1] => 19
[3,4,1,2,5] => 1
[3,4,1,5,2] => 5
[3,4,2,1,5] => 2
[3,4,2,5,1] => 7
[3,4,5,1,2] => 5
[3,4,5,2,1] => 12
[3,5,1,2,4] => 5
[3,5,1,4,2] => 21
[3,5,2,1,4] => 16
[3,5,2,4,1] => 46
[3,5,4,1,2] => 26
[3,5,4,2,1] => 89
[4,1,2,3,5] => 0
[4,1,2,5,3] => 2
[4,1,3,2,5] => 1
[4,1,3,5,2] => 4
[4,1,5,2,3] => 5
[4,1,5,3,2] => 16
[4,2,1,3,5] => 1
[4,2,1,5,3] => 11
[4,2,3,1,5] => 4
[4,2,3,5,1] => 9
[4,2,5,1,3] => 21
[4,2,5,3,1] => 46
[4,3,1,2,5] => 2
[4,3,1,5,2] => 16
[4,3,2,1,5] => 10
[4,3,2,5,1] => 37
[4,3,5,1,2] => 26
[4,3,5,2,1] => 89
[4,5,1,2,3] => 5
[4,5,1,3,2] => 26
[4,5,2,1,3] => 26
[4,5,2,3,1] => 88
[4,5,3,1,2] => 52
[4,5,3,2,1] => 250
[5,1,2,3,4] => 0
[5,1,2,4,3] => 2
[5,1,3,2,4] => 2
[5,1,3,4,2] => 9
[5,1,4,2,3] => 7
[5,1,4,3,2] => 37
[5,2,1,3,4] => 2
[5,2,1,4,3] => 19
[5,2,3,1,4] => 9
[5,2,3,4,1] => 24
[5,2,4,1,3] => 46
[5,2,4,3,1] => 133
[5,3,1,2,4] => 7
[5,3,1,4,2] => 46
[5,3,2,1,4] => 37
[5,3,2,4,1] => 133
[5,3,4,1,2] => 88
[5,3,4,2,1] => 366
[5,4,1,2,3] => 12
[5,4,1,3,2] => 89
[5,4,2,1,3] => 89
[5,4,2,3,1] => 366
[5,4,3,1,2] => 250
[5,4,3,2,1] => 1386
[1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => 0
[1,2,3,5,4,6] => 0
[1,2,3,5,6,4] => 0
[1,2,3,6,4,5] => 0
[1,2,3,6,5,4] => 0
[1,2,4,3,5,6] => 0
[1,2,4,3,6,5] => 1
[1,2,4,5,3,6] => 0
[1,2,4,5,6,3] => 0
[1,2,4,6,3,5] => 1
[1,2,4,6,5,3] => 1
[1,2,5,3,4,6] => 0
[1,2,5,3,6,4] => 1
[1,2,5,4,3,6] => 0
[1,2,5,4,6,3] => 1
[1,2,5,6,3,4] => 1
[1,2,5,6,4,3] => 2
[1,2,6,3,4,5] => 0
[1,2,6,3,5,4] => 1
[1,2,6,4,3,5] => 1
[1,2,6,4,5,3] => 4
[1,2,6,5,3,4] => 2
[1,2,6,5,4,3] => 10
[1,3,2,4,5,6] => 0
[1,3,2,4,6,5] => 1
[1,3,2,5,4,6] => 1
[1,3,2,5,6,4] => 2
[1,3,2,6,4,5] => 2
[1,3,2,6,5,4] => 6
[1,3,4,2,5,6] => 0
[1,3,4,2,6,5] => 2
[1,3,4,5,2,6] => 0
[1,3,4,5,6,2] => 0
[1,3,4,6,2,5] => 2
[1,3,4,6,5,2] => 2
[1,3,5,2,4,6] => 1
[1,3,5,2,6,4] => 5
[1,3,5,4,2,6] => 1
[1,3,5,4,6,2] => 2
[1,3,5,6,2,4] => 5
[1,3,5,6,4,2] => 7
[1,3,6,2,4,5] => 2
[1,3,6,2,5,4] => 11
[1,3,6,4,2,5] => 4
[1,3,6,4,5,2] => 9
[1,3,6,5,2,4] => 16
[1,3,6,5,4,2] => 37
[1,4,2,3,5,6] => 0
[1,4,2,3,6,5] => 2
[1,4,2,5,3,6] => 1
[1,4,2,5,6,3] => 2
[1,4,2,6,3,5] => 5
[1,4,2,6,5,3] => 11
[1,4,3,2,5,6] => 0
[1,4,3,2,6,5] => 6
[1,4,3,5,2,6] => 1
[1,4,3,5,6,2] => 2
[1,4,3,6,2,5] => 11
[1,4,3,6,5,2] => 19
[1,4,5,2,3,6] => 1
[1,4,5,2,6,3] => 5
[1,4,5,3,2,6] => 2
[1,4,5,3,6,2] => 7
[1,4,5,6,2,3] => 5
[1,4,5,6,3,2] => 12
[1,4,6,2,3,5] => 5
[1,4,6,2,5,3] => 21
[1,4,6,3,2,5] => 16
[1,4,6,3,5,2] => 46
[1,4,6,5,2,3] => 26
[1,4,6,5,3,2] => 89
[1,5,2,3,4,6] => 0
[1,5,2,3,6,4] => 2
[1,5,2,4,3,6] => 1
[1,5,2,4,6,3] => 4
[1,5,2,6,3,4] => 5
[1,5,2,6,4,3] => 16
[1,5,3,2,4,6] => 1
[1,5,3,2,6,4] => 11
[1,5,3,4,2,6] => 4
[1,5,3,4,6,2] => 9
[1,5,3,6,2,4] => 21
[1,5,3,6,4,2] => 46
[1,5,4,2,3,6] => 2
[1,5,4,2,6,3] => 16
[1,5,4,3,2,6] => 10
[1,5,4,3,6,2] => 37
[1,5,4,6,2,3] => 26
[1,5,4,6,3,2] => 89
[1,5,6,2,3,4] => 5
[1,5,6,2,4,3] => 26
[1,5,6,3,2,4] => 26
[1,5,6,3,4,2] => 88
[1,5,6,4,2,3] => 52
[1,5,6,4,3,2] => 250
[1,6,2,3,4,5] => 0
[1,6,2,3,5,4] => 2
[1,6,2,4,3,5] => 2
[1,6,2,4,5,3] => 9
[1,6,2,5,3,4] => 7
[1,6,2,5,4,3] => 37
[1,6,3,2,4,5] => 2
[1,6,3,2,5,4] => 19
[1,6,3,4,2,5] => 9
[1,6,3,4,5,2] => 24
[1,6,3,5,2,4] => 46
[1,6,3,5,4,2] => 133
[1,6,4,2,3,5] => 7
[1,6,4,2,5,3] => 46
[1,6,4,3,2,5] => 37
[1,6,4,3,5,2] => 133
[1,6,4,5,2,3] => 88
[1,6,4,5,3,2] => 366
[1,6,5,2,3,4] => 12
[1,6,5,2,4,3] => 89
[1,6,5,3,2,4] => 89
[1,6,5,3,4,2] => 366
[1,6,5,4,2,3] => 250
[1,6,5,4,3,2] => 1386
[2,1,3,4,5,6] => 0
[2,1,3,4,6,5] => 1
[2,1,3,5,4,6] => 1
[2,1,3,5,6,4] => 2
[2,1,3,6,4,5] => 2
[2,1,3,6,5,4] => 6
[2,1,4,3,5,6] => 1
[2,1,4,3,6,5] => 6
[2,1,4,5,3,6] => 2
[2,1,4,5,6,3] => 3
[2,1,4,6,3,5] => 9
[2,1,4,6,5,3] => 16
[2,1,5,3,4,6] => 2
[2,1,5,3,6,4] => 9
[2,1,5,4,3,6] => 6
[2,1,5,4,6,3] => 16
[2,1,5,6,3,4] => 12
[2,1,5,6,4,3] => 35
[2,1,6,3,4,5] => 3
[2,1,6,3,5,4] => 16
[2,1,6,4,3,5] => 16
[2,1,6,4,5,3] => 50
[2,1,6,5,3,4] => 35
[2,1,6,5,4,3] => 156
[2,3,1,4,5,6] => 0
[2,3,1,4,6,5] => 2
[2,3,1,5,4,6] => 2
[2,3,1,5,6,4] => 6
[2,3,1,6,4,5] => 6
[2,3,1,6,5,4] => 24
[2,3,4,1,5,6] => 0
[2,3,4,1,6,5] => 3
[2,3,4,5,1,6] => 0
[2,3,4,5,6,1] => 0
[2,3,4,6,1,5] => 3
[2,3,4,6,5,1] => 3
[2,3,5,1,4,6] => 2
[2,3,5,1,6,4] => 11
[2,3,5,4,1,6] => 2
[2,3,5,4,6,1] => 3
[2,3,5,6,1,4] => 11
[2,3,5,6,4,1] => 14
[2,3,6,1,4,5] => 6
[2,3,6,1,5,4] => 36
[2,3,6,4,1,5] => 9
[2,3,6,4,5,1] => 16
[2,3,6,5,1,4] => 47
[2,3,6,5,4,1] => 86
[2,4,1,3,5,6] => 1
[2,4,1,3,6,5] => 9
[2,4,1,5,3,6] => 5
[2,4,1,5,6,3] => 11
[2,4,1,6,3,5] => 24
[2,4,1,6,5,3] => 63
[2,4,3,1,5,6] => 1
[2,4,3,1,6,5] => 16
[2,4,3,5,1,6] => 2
[2,4,3,5,6,1] => 3
[2,4,3,6,1,5] => 23
[2,4,3,6,5,1] => 34
[2,4,5,1,3,6] => 5
[2,4,5,1,6,3] => 21
[2,4,5,3,1,6] => 7
[2,4,5,3,6,1] => 14
[2,4,5,6,1,3] => 21
[2,4,5,6,3,1] => 35
[2,4,6,1,3,5] => 24
[2,4,6,1,5,3] => 103
[2,4,6,3,1,5] => 47
[2,4,6,3,5,1] => 100
[2,4,6,5,1,3] => 124
[2,4,6,5,3,1] => 273
[2,5,1,3,4,6] => 2
[2,5,1,3,6,4] => 14
[2,5,1,4,3,6] => 11
[2,5,1,4,6,3] => 35
[2,5,1,6,3,4] => 34
[2,5,1,6,4,3] => 129
[2,5,3,1,4,6] => 4
[2,5,3,1,6,4] => 35
[2,5,3,4,1,6] => 9
[2,5,3,4,6,1] => 16
[2,5,3,6,1,4] => 55
[2,5,3,6,4,1] => 99
[2,5,4,1,3,6] => 16
[2,5,4,1,6,3] => 88
[2,5,4,3,1,6] => 37
[2,5,4,3,6,1] => 86
[2,5,4,6,1,3] => 123
[2,5,4,6,3,1] => 272
[2,5,6,1,3,4] => 34
[2,5,6,1,4,3] => 184
[2,5,6,3,1,4] => 89
[2,5,6,3,4,1] => 223
[2,5,6,4,1,3] => 307
[2,5,6,4,3,1] => 872
[2,6,1,3,4,5] => 3
[2,6,1,3,5,4] => 23
[2,6,1,4,3,5] => 23
[2,6,1,4,5,3] => 88
[2,6,1,5,3,4] => 73
[2,6,1,5,4,3] => 383
[2,6,3,1,4,5] => 9
[2,6,3,1,5,4] => 78
[2,6,3,4,1,5] => 24
[2,6,3,4,5,1] => 50
[2,6,3,5,1,4] => 151
[2,6,3,5,4,1] => 337
[2,6,4,1,3,5] => 47
[2,6,4,1,5,3] => 273
[2,6,4,3,1,5] => 134
[2,6,4,3,5,1] => 339
[2,6,4,5,1,3] => 453
[2,6,4,5,3,1] => 1199
[2,6,5,1,3,4] => 107
[2,6,5,1,4,3] => 744
[2,6,5,3,1,4] => 368
[2,6,5,3,4,1] => 1074
[2,6,5,4,1,3] => 1602
[2,6,5,4,3,1] => 5251
[3,1,2,4,5,6] => 0
[3,1,2,4,6,5] => 2
[3,1,2,5,4,6] => 2
[3,1,2,5,6,4] => 6
[3,1,2,6,4,5] => 6
[3,1,2,6,5,4] => 24
[3,1,4,2,5,6] => 1
[3,1,4,2,6,5] => 9
[3,1,4,5,2,6] => 2
[3,1,4,5,6,2] => 3
[3,1,4,6,2,5] => 14
[3,1,4,6,5,2] => 23
[3,1,5,2,4,6] => 5
[3,1,5,2,6,4] => 24
[3,1,5,4,2,6] => 11
[3,1,5,4,6,2] => 23
[3,1,5,6,2,4] => 34
[3,1,5,6,4,2] => 73
[3,1,6,2,4,5] => 11
[3,1,6,2,5,4] => 63
[3,1,6,4,2,5] => 35
[3,1,6,4,5,2] => 88
[3,1,6,5,2,4] => 129
[3,1,6,5,4,2] => 383
[3,2,1,4,5,6] => 0
[3,2,1,4,6,5] => 6
[3,2,1,5,4,6] => 6
[3,2,1,5,6,4] => 24
[3,2,1,6,4,5] => 24
[3,2,1,6,5,4] => 120
[3,2,4,1,5,6] => 1
[3,2,4,1,6,5] => 16
[3,2,4,5,1,6] => 2
[3,2,4,5,6,1] => 3
[3,2,4,6,1,5] => 23
[3,2,4,6,5,1] => 34
[3,2,5,1,4,6] => 11
[3,2,5,1,6,4] => 63
[3,2,5,4,1,6] => 19
[3,2,5,4,6,1] => 34
[3,2,5,6,1,4] => 83
[3,2,5,6,4,1] => 145
[3,2,6,1,4,5] => 36
[3,2,6,1,5,4] => 238
[3,2,6,4,1,5] => 78
[3,2,6,4,5,1] => 162
[3,2,6,5,1,4] => 403
[3,2,6,5,4,1] => 909
[3,4,1,2,5,6] => 1
[3,4,1,2,6,5] => 12
[3,4,1,5,2,6] => 5
[3,4,1,5,6,2] => 11
[3,4,1,6,2,5] => 34
[3,4,1,6,5,2] => 83
[3,4,2,1,5,6] => 2
[3,4,2,1,6,5] => 35
[3,4,2,5,1,6] => 7
[3,4,2,5,6,1] => 14
[3,4,2,6,1,5] => 73
[3,4,2,6,5,1] => 145
[3,4,5,1,2,6] => 5
[3,4,5,1,6,2] => 21
[3,4,5,2,1,6] => 12
[3,4,5,2,6,1] => 35
[3,4,5,6,1,2] => 21
[3,4,5,6,2,1] => 56
[3,4,6,1,2,5] => 34
[3,4,6,1,5,2] => 138
[3,4,6,2,1,5] => 107
[3,4,6,2,5,1] => 301
[3,4,6,5,1,2] => 159
[3,4,6,5,2,1] => 530
[3,5,1,2,4,6] => 5
[3,5,1,2,6,4] => 34
[3,5,1,4,2,6] => 21
[3,5,1,4,6,2] => 55
[3,5,1,6,2,4] => 84
[3,5,1,6,4,2] => 252
[3,5,2,1,4,6] => 16
[3,5,2,1,6,4] => 129
[3,5,2,4,1,6] => 46
[3,5,2,4,6,1] => 99
[3,5,2,6,1,4] => 252
[3,5,2,6,4,1] => 574
[3,5,4,1,2,6] => 26
[3,5,4,1,6,2] => 123
[3,5,4,2,1,6] => 89
[3,5,4,2,6,1] => 272
[3,5,4,6,1,2] => 158
[3,5,4,6,2,1] => 528
[3,5,6,1,2,4] => 84
[3,5,6,1,4,2] => 378
[3,5,6,2,1,4] => 336
[3,5,6,2,4,1] => 1036
[3,5,6,4,1,2] => 536
[3,5,6,4,2,1] => 2184
[3,6,1,2,4,5] => 11
[3,6,1,2,5,4] => 83
[3,6,1,4,2,5] => 55
[3,6,1,4,5,2] => 169
[3,6,1,5,2,4] => 252
[3,6,1,5,4,2] => 936
[3,6,2,1,4,5] => 47
[3,6,2,1,5,4] => 403
[3,6,2,4,1,5] => 151
[3,6,2,4,5,1] => 368
[3,6,2,5,1,4] => 938
[3,6,2,5,4,1] => 2530
click to show generating function       
Description
The number of short braid edges in the graph of braid moves of a permutation.
Given a permutation $\pi$, let $\operatorname{Red}(\pi)$ denote the set of reduced words for $\pi$ in terms of simple transpositions $s_i = (i,i+1)$. We now say that two reduced words are connected by a short braid move if they are obtained from each other by a modification of the form $s_i s_j \leftrightarrow s_j s_i$ for $|i-j| > 1$ as a consecutive subword of a reduced word.
For example, the two reduced words $s_1s_3s_2$ and $s_3s_1s_2$ for
$$(1243) = (12)(34)(23) = (34)(12)(23)$$
share an edge because they are obtained from each other by interchanging $s_1s_3 \leftrightarrow s_3s_1$.
This statistic counts the number of such short braid moves among all reduced words.
Code
def short_braid_move_graph(pi):
    V = [ tuple(w) for w in pi.reduced_words() ]
    is_edge = lambda w1,w2: w1 != w2 and any( w1[:i] == w2[:i] and w1[i+2:] == w2[i+2:] for i in range(len(w1)-1) )
    return Graph([V,is_edge])

def statistic(pi):
    return len( short_braid_move_graph(pi).edges() )
Created
Jul 03, 2017 at 16:58 by Christian Stump
Updated
Jul 03, 2017 at 16:58 by Christian Stump