Identifier
- St000881: Permutations ⟶ ℤ
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
[3,6,4,1,2,5]=>89
[3,6,4,1,5,2]=>452
[3,6,4,2,1,5]=>368
[3,6,4,2,5,1]=>1197
[3,6,4,5,1,2]=>681
[3,6,4,5,2,1]=>2654
[3,6,5,1,2,4]=>336
[3,6,5,1,4,2]=>1818
[3,6,5,2,1,4]=>1652
[3,6,5,2,4,1]=>5818
[3,6,5,4,1,2]=>3203
[3,6,5,4,2,1]=>14861
[4,1,2,3,5,6]=>0
[4,1,2,3,6,5]=>3
[4,1,2,5,3,6]=>2
[4,1,2,5,6,3]=>6
[4,1,2,6,3,5]=>11
[4,1,2,6,5,3]=>36
[4,1,3,2,5,6]=>1
[4,1,3,2,6,5]=>16
[4,1,3,5,2,6]=>4
[4,1,3,5,6,2]=>9
[4,1,3,6,2,5]=>35
[4,1,3,6,5,2]=>78
[4,1,5,2,3,6]=>5
[4,1,5,2,6,3]=>24
[4,1,5,3,2,6]=>16
[4,1,5,3,6,2]=>47
[4,1,5,6,2,3]=>34
[4,1,5,6,3,2]=>107
[4,1,6,2,3,5]=>21
[4,1,6,2,5,3]=>103
[4,1,6,3,2,5]=>88
[4,1,6,3,5,2]=>273
[4,1,6,5,2,3]=>184
[4,1,6,5,3,2]=>744
[4,2,1,3,5,6]=>1
[4,2,1,3,6,5]=>16
[4,2,1,5,3,6]=>11
[4,2,1,5,6,3]=>36
[4,2,1,6,3,5]=>63
[4,2,1,6,5,3]=>238
[4,2,3,1,5,6]=>4
[4,2,3,1,6,5]=>50
[4,2,3,5,1,6]=>9
[4,2,3,5,6,1]=>16
[4,2,3,6,1,5]=>88
[4,2,3,6,5,1]=>162
[4,2,5,1,3,6]=>21
[4,2,5,1,6,3]=>103
[4,2,5,3,1,6]=>46
[4,2,5,3,6,1]=>100
[4,2,5,6,1,3]=>138
[4,2,5,6,3,1]=>301
[4,2,6,1,3,5]=>103
[4,2,6,1,5,3]=>540
[4,2,6,3,1,5]=>273
[4,2,6,3,5,1]=>654
[4,2,6,5,1,3]=>872
[4,2,6,5,3,1]=>2332
[4,3,1,2,5,6]=>2
[4,3,1,2,6,5]=>35
[4,3,1,5,2,6]=>16
[4,3,1,5,6,2]=>47
[4,3,1,6,2,5]=>129
[4,3,1,6,5,2]=>403
[4,3,2,1,5,6]=>10
[4,3,2,1,6,5]=>156
[4,3,2,5,1,6]=>37
[4,3,2,5,6,1]=>86
[4,3,2,6,1,5]=>383
[4,3,2,6,5,1]=>909
[4,3,5,1,2,6]=>26
[4,3,5,1,6,2]=>124
[4,3,5,2,1,6]=>89
[4,3,5,2,6,1]=>273
[4,3,5,6,1,2]=>159
[4,3,5,6,2,1]=>530
[4,3,6,1,2,5]=>184
[4,3,6,1,5,2]=>872
[4,3,6,2,1,5]=>744
[4,3,6,2,5,1]=>2332
[4,3,6,5,1,2]=>1274
[4,3,6,5,2,1]=>5038
[4,5,1,2,3,6]=>5
[4,5,1,2,6,3]=>34
[4,5,1,3,2,6]=>26
[4,5,1,3,6,2]=>89
[4,5,1,6,2,3]=>84
[4,5,1,6,3,2]=>336
[4,5,2,1,3,6]=>26
[4,5,2,1,6,3]=>184
[4,5,2,3,1,6]=>88
[4,5,2,3,6,1]=>223
[4,5,2,6,1,3]=>378
[4,5,2,6,3,1]=>1036
[4,5,3,1,2,6]=>52
[4,5,3,1,6,2]=>307
[4,5,3,2,1,6]=>250
[4,5,3,2,6,1]=>872
[4,5,3,6,1,2]=>536
[4,5,3,6,2,1]=>2184
[4,5,6,1,2,3]=>84
[4,5,6,1,3,2]=>462
[4,5,6,2,1,3]=>462
[4,5,6,2,3,1]=>1666
[4,5,6,3,1,2]=>998
[4,5,6,3,2,1]=>5100
[4,6,1,2,3,5]=>21
[4,6,1,2,5,3]=>138
[4,6,1,3,2,5]=>123
[4,6,1,3,5,2]=>452
[4,6,1,5,2,3]=>378
[4,6,1,5,3,2]=>1818
[4,6,2,1,3,5]=>124
[4,6,2,1,5,3]=>872
[4,6,2,3,1,5]=>453
[4,6,2,3,5,1]=>1253
[4,6,2,5,1,3]=>1988
[4,6,2,5,3,1]=>6266
[4,6,3,1,2,5]=>307
[4,6,3,1,5,2]=>1799
[4,6,3,2,1,5]=>1602
[4,6,3,2,5,1]=>5787
[4,6,3,5,1,2]=>3457
[4,6,3,5,2,1]=>15851
[4,6,5,1,2,3]=>462
[4,6,5,1,3,2]=>2952
[4,6,5,2,1,3]=>2954
[4,6,5,2,3,1]=>11864
[4,6,5,3,1,2]=>7617
[4,6,5,3,2,1]=>42874
[5,1,2,3,4,6]=>0
[5,1,2,3,6,4]=>3
[5,1,2,4,3,6]=>2
[5,1,2,4,6,3]=>9
[5,1,2,6,3,4]=>11
[5,1,2,6,4,3]=>47
[5,1,3,2,4,6]=>2
[5,1,3,2,6,4]=>23
[5,1,3,4,2,6]=>9
[5,1,3,4,6,2]=>24
[5,1,3,6,2,4]=>55
[5,1,3,6,4,2]=>151
[5,1,4,2,3,6]=>7
[5,1,4,2,6,3]=>47
[5,1,4,3,2,6]=>37
[5,1,4,3,6,2]=>134
[5,1,4,6,2,3]=>89
[5,1,4,6,3,2]=>368
[5,1,6,2,3,4]=>21
[5,1,6,2,4,3]=>124
[5,1,6,3,2,4]=>123
[5,1,6,3,4,2]=>453
[5,1,6,4,2,3]=>307
[5,1,6,4,3,2]=>1602
[5,2,1,3,4,6]=>2
[5,2,1,3,6,4]=>23
[5,2,1,4,3,6]=>19
[5,2,1,4,6,3]=>78
[5,2,1,6,3,4]=>83
[5,2,1,6,4,3]=>403
[5,2,3,1,4,6]=>9
[5,2,3,1,6,4]=>88
[5,2,3,4,1,6]=>24
[5,2,3,4,6,1]=>50
[5,2,3,6,1,4]=>169
[5,2,3,6,4,1]=>368
[5,2,4,1,3,6]=>46
[5,2,4,1,6,3]=>273
[5,2,4,3,1,6]=>133
[5,2,4,3,6,1]=>339
[5,2,4,6,1,3]=>452
[5,2,4,6,3,1]=>1197
[5,2,6,1,3,4]=>138
[5,2,6,1,4,3]=>872
[5,2,6,3,1,4]=>452
[5,2,6,3,4,1]=>1253
[5,2,6,4,1,3]=>1799
[5,2,6,4,3,1]=>5787
[5,3,1,2,4,6]=>7
[5,3,1,2,6,4]=>73
[5,3,1,4,2,6]=>46
[5,3,1,4,6,2]=>151
[5,3,1,6,2,4]=>252
[5,3,1,6,4,2]=>938
[5,3,2,1,4,6]=>37
[5,3,2,1,6,4]=>383
[5,3,2,4,1,6]=>133
[5,3,2,4,6,1]=>337
[5,3,2,6,1,4]=>936
[5,3,2,6,4,1]=>2530
[5,3,4,1,2,6]=>88
[5,3,4,1,6,2]=>453
[5,3,4,2,1,6]=>366
[5,3,4,2,6,1]=>1199
[5,3,4,6,1,2]=>681
[5,3,4,6,2,1]=>2654
[5,3,6,1,2,4]=>378
[5,3,6,1,4,2]=>1988
[5,3,6,2,1,4]=>1818
[5,3,6,2,4,1]=>6266
[5,3,6,4,1,2]=>3457
[5,3,6,4,2,1]=>15851
[5,4,1,2,3,6]=>12
[5,4,1,2,6,3]=>107
[5,4,1,3,2,6]=>89
[5,4,1,3,6,2]=>368
[5,4,1,6,2,3]=>336
[5,4,1,6,3,2]=>1652
[5,4,2,1,3,6]=>89
[5,4,2,1,6,3]=>744
[5,4,2,3,1,6]=>366
[5,4,2,3,6,1]=>1074
[5,4,2,6,1,3]=>1818
[5,4,2,6,3,1]=>5818
[5,4,3,1,2,6]=>250
[5,4,3,1,6,2]=>1602
[5,4,3,2,1,6]=>1386
[5,4,3,2,6,1]=>5251
[5,4,3,6,1,2]=>3203
[5,4,3,6,2,1]=>14861
[5,4,6,1,2,3]=>462
[5,4,6,1,3,2]=>2954
[5,4,6,2,1,3]=>2952
[5,4,6,2,3,1]=>11864
[5,4,6,3,1,2]=>7617
[5,4,6,3,2,1]=>42874
[5,6,1,2,3,4]=>21
[5,6,1,2,4,3]=>159
[5,6,1,3,2,4]=>158
[5,6,1,3,4,2]=>681
[5,6,1,4,2,3]=>536
[5,6,1,4,3,2]=>3203
[5,6,2,1,3,4]=>159
[5,6,2,1,4,3]=>1274
[5,6,2,3,1,4]=>681
[5,6,2,3,4,1]=>2150
[5,6,2,4,1,3]=>3457
[5,6,2,4,3,1]=>12687
[5,6,3,1,2,4]=>536
[5,6,3,1,4,2]=>3457
[5,6,3,2,1,4]=>3203
[5,6,3,2,4,1]=>12687
[5,6,3,4,1,2]=>7682
[5,6,3,4,2,1]=>39850
[5,6,4,1,2,3]=>998
[5,6,4,1,3,2]=>7617
[5,6,4,2,1,3]=>7617
[5,6,4,2,3,1]=>34676
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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.
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
Mar 12, 2026 at 17:41 by Nupur Jain
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