Identifier
- St000880: Permutations ⟶ ℤ
Values
=>
[1,2]=>1
[2,1]=>1
[1,2,3]=>1
[1,3,2]=>1
[2,1,3]=>1
[2,3,1]=>1
[3,1,2]=>1
[3,2,1]=>1
[1,2,3,4]=>1
[1,2,4,3]=>1
[1,3,2,4]=>1
[1,3,4,2]=>1
[1,4,2,3]=>1
[1,4,3,2]=>1
[2,1,3,4]=>1
[2,1,4,3]=>2
[2,3,1,4]=>1
[2,3,4,1]=>1
[2,4,1,3]=>2
[2,4,3,1]=>2
[3,1,2,4]=>1
[3,1,4,2]=>2
[3,2,1,4]=>1
[3,2,4,1]=>2
[3,4,1,2]=>2
[3,4,2,1]=>3
[4,1,2,3]=>1
[4,1,3,2]=>2
[4,2,1,3]=>2
[4,2,3,1]=>4
[4,3,1,2]=>3
[4,3,2,1]=>8
[1,2,3,4,5]=>1
[1,2,3,5,4]=>1
[1,2,4,3,5]=>1
[1,2,4,5,3]=>1
[1,2,5,3,4]=>1
[1,2,5,4,3]=>1
[1,3,2,4,5]=>1
[1,3,2,5,4]=>2
[1,3,4,2,5]=>1
[1,3,4,5,2]=>1
[1,3,5,2,4]=>2
[1,3,5,4,2]=>2
[1,4,2,3,5]=>1
[1,4,2,5,3]=>2
[1,4,3,2,5]=>1
[1,4,3,5,2]=>2
[1,4,5,2,3]=>2
[1,4,5,3,2]=>3
[1,5,2,3,4]=>1
[1,5,2,4,3]=>2
[1,5,3,2,4]=>2
[1,5,3,4,2]=>4
[1,5,4,2,3]=>3
[1,5,4,3,2]=>8
[2,1,3,4,5]=>1
[2,1,3,5,4]=>2
[2,1,4,3,5]=>2
[2,1,4,5,3]=>3
[2,1,5,3,4]=>3
[2,1,5,4,3]=>6
[2,3,1,4,5]=>1
[2,3,1,5,4]=>3
[2,3,4,1,5]=>1
[2,3,4,5,1]=>1
[2,3,5,1,4]=>3
[2,3,5,4,1]=>3
[2,4,1,3,5]=>2
[2,4,1,5,3]=>5
[2,4,3,1,5]=>2
[2,4,3,5,1]=>3
[2,4,5,1,3]=>5
[2,4,5,3,1]=>6
[2,5,1,3,4]=>3
[2,5,1,4,3]=>9
[2,5,3,1,4]=>4
[2,5,3,4,1]=>7
[2,5,4,1,3]=>11
[2,5,4,3,1]=>20
[3,1,2,4,5]=>1
[3,1,2,5,4]=>3
[3,1,4,2,5]=>2
[3,1,4,5,2]=>3
[3,1,5,2,4]=>5
[3,1,5,4,2]=>9
[3,2,1,4,5]=>1
[3,2,1,5,4]=>6
[3,2,4,1,5]=>2
[3,2,4,5,1]=>3
[3,2,5,1,4]=>9
[3,2,5,4,1]=>14
[3,4,1,2,5]=>2
[3,4,1,5,2]=>5
[3,4,2,1,5]=>3
[3,4,2,5,1]=>6
[3,4,5,1,2]=>5
[3,4,5,2,1]=>9
[3,5,1,2,4]=>5
[3,5,1,4,2]=>14
[3,5,2,1,4]=>11
[3,5,2,4,1]=>25
[3,5,4,1,2]=>16
[3,5,4,2,1]=>42
[4,1,2,3,5]=>1
[4,1,2,5,3]=>3
[4,1,3,2,5]=>2
[4,1,3,5,2]=>4
[4,1,5,2,3]=>5
[4,1,5,3,2]=>11
[4,2,1,3,5]=>2
[4,2,1,5,3]=>9
[4,2,3,1,5]=>4
[4,2,3,5,1]=>7
[4,2,5,1,3]=>14
[4,2,5,3,1]=>25
[4,3,1,2,5]=>3
[4,3,1,5,2]=>11
[4,3,2,1,5]=>8
[4,3,2,5,1]=>20
[4,3,5,1,2]=>16
[4,3,5,2,1]=>42
[4,5,1,2,3]=>5
[4,5,1,3,2]=>16
[4,5,2,1,3]=>16
[4,5,2,3,1]=>41
[4,5,3,1,2]=>28
[4,5,3,2,1]=>99
[5,1,2,3,4]=>1
[5,1,2,4,3]=>3
[5,1,3,2,4]=>3
[5,1,3,4,2]=>7
[5,1,4,2,3]=>6
[5,1,4,3,2]=>20
[5,2,1,3,4]=>3
[5,2,1,4,3]=>14
[5,2,3,1,4]=>7
[5,2,3,4,1]=>14
[5,2,4,1,3]=>25
[5,2,4,3,1]=>56
[5,3,1,2,4]=>6
[5,3,1,4,2]=>25
[5,3,2,1,4]=>20
[5,3,2,4,1]=>56
[5,3,4,1,2]=>41
[5,3,4,2,1]=>133
[5,4,1,2,3]=>9
[5,4,1,3,2]=>42
[5,4,2,1,3]=>42
[5,4,2,3,1]=>133
[5,4,3,1,2]=>99
[5,4,3,2,1]=>432
[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]=>1
[1,2,3,6,4,5]=>1
[1,2,3,6,5,4]=>1
[1,2,4,3,5,6]=>1
[1,2,4,3,6,5]=>2
[1,2,4,5,3,6]=>1
[1,2,4,5,6,3]=>1
[1,2,4,6,3,5]=>2
[1,2,4,6,5,3]=>2
[1,2,5,3,4,6]=>1
[1,2,5,3,6,4]=>2
[1,2,5,4,3,6]=>1
[1,2,5,4,6,3]=>2
[1,2,5,6,3,4]=>2
[1,2,5,6,4,3]=>3
[1,2,6,3,4,5]=>1
[1,2,6,3,5,4]=>2
[1,2,6,4,3,5]=>2
[1,2,6,4,5,3]=>4
[1,2,6,5,3,4]=>3
[1,2,6,5,4,3]=>8
[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]=>3
[1,3,2,6,4,5]=>3
[1,3,2,6,5,4]=>6
[1,3,4,2,5,6]=>1
[1,3,4,2,6,5]=>3
[1,3,4,5,2,6]=>1
[1,3,4,5,6,2]=>1
[1,3,4,6,2,5]=>3
[1,3,4,6,5,2]=>3
[1,3,5,2,4,6]=>2
[1,3,5,2,6,4]=>5
[1,3,5,4,2,6]=>2
[1,3,5,4,6,2]=>3
[1,3,5,6,2,4]=>5
[1,3,5,6,4,2]=>6
[1,3,6,2,4,5]=>3
[1,3,6,2,5,4]=>9
[1,3,6,4,2,5]=>4
[1,3,6,4,5,2]=>7
[1,3,6,5,2,4]=>11
[1,3,6,5,4,2]=>20
[1,4,2,3,5,6]=>1
[1,4,2,3,6,5]=>3
[1,4,2,5,3,6]=>2
[1,4,2,5,6,3]=>3
[1,4,2,6,3,5]=>5
[1,4,2,6,5,3]=>9
[1,4,3,2,5,6]=>1
[1,4,3,2,6,5]=>6
[1,4,3,5,2,6]=>2
[1,4,3,5,6,2]=>3
[1,4,3,6,2,5]=>9
[1,4,3,6,5,2]=>14
[1,4,5,2,3,6]=>2
[1,4,5,2,6,3]=>5
[1,4,5,3,2,6]=>3
[1,4,5,3,6,2]=>6
[1,4,5,6,2,3]=>5
[1,4,5,6,3,2]=>9
[1,4,6,2,3,5]=>5
[1,4,6,2,5,3]=>14
[1,4,6,3,2,5]=>11
[1,4,6,3,5,2]=>25
[1,4,6,5,2,3]=>16
[1,4,6,5,3,2]=>42
[1,5,2,3,4,6]=>1
[1,5,2,3,6,4]=>3
[1,5,2,4,3,6]=>2
[1,5,2,4,6,3]=>4
[1,5,2,6,3,4]=>5
[1,5,2,6,4,3]=>11
[1,5,3,2,4,6]=>2
[1,5,3,2,6,4]=>9
[1,5,3,4,2,6]=>4
[1,5,3,4,6,2]=>7
[1,5,3,6,2,4]=>14
[1,5,3,6,4,2]=>25
[1,5,4,2,3,6]=>3
[1,5,4,2,6,3]=>11
[1,5,4,3,2,6]=>8
[1,5,4,3,6,2]=>20
[1,5,4,6,2,3]=>16
[1,5,4,6,3,2]=>42
[1,5,6,2,3,4]=>5
[1,5,6,2,4,3]=>16
[1,5,6,3,2,4]=>16
[1,5,6,3,4,2]=>41
[1,5,6,4,2,3]=>28
[1,5,6,4,3,2]=>99
[1,6,2,3,4,5]=>1
[1,6,2,3,5,4]=>3
[1,6,2,4,3,5]=>3
[1,6,2,4,5,3]=>7
[1,6,2,5,3,4]=>6
[1,6,2,5,4,3]=>20
[1,6,3,2,4,5]=>3
[1,6,3,2,5,4]=>14
[1,6,3,4,2,5]=>7
[1,6,3,4,5,2]=>14
[1,6,3,5,2,4]=>25
[1,6,3,5,4,2]=>56
[1,6,4,2,3,5]=>6
[1,6,4,2,5,3]=>25
[1,6,4,3,2,5]=>20
[1,6,4,3,5,2]=>56
[1,6,4,5,2,3]=>41
[1,6,4,5,3,2]=>133
[1,6,5,2,3,4]=>9
[1,6,5,2,4,3]=>42
[1,6,5,3,2,4]=>42
[1,6,5,3,4,2]=>133
[1,6,5,4,2,3]=>99
[1,6,5,4,3,2]=>432
[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]=>3
[2,1,3,6,4,5]=>3
[2,1,3,6,5,4]=>6
[2,1,4,3,5,6]=>2
[2,1,4,3,6,5]=>6
[2,1,4,5,3,6]=>3
[2,1,4,5,6,3]=>4
[2,1,4,6,3,5]=>8
[2,1,4,6,5,3]=>12
[2,1,5,3,4,6]=>3
[2,1,5,3,6,4]=>8
[2,1,5,4,3,6]=>6
[2,1,5,4,6,3]=>12
[2,1,5,6,3,4]=>10
[2,1,5,6,4,3]=>22
[2,1,6,3,4,5]=>4
[2,1,6,3,5,4]=>12
[2,1,6,4,3,5]=>12
[2,1,6,4,5,3]=>28
[2,1,6,5,3,4]=>22
[2,1,6,5,4,3]=>72
[2,3,1,4,5,6]=>1
[2,3,1,4,6,5]=>3
[2,3,1,5,4,6]=>3
[2,3,1,5,6,4]=>6
[2,3,1,6,4,5]=>6
[2,3,1,6,5,4]=>17
[2,3,4,1,5,6]=>1
[2,3,4,1,6,5]=>4
[2,3,4,5,1,6]=>1
[2,3,4,5,6,1]=>1
[2,3,4,6,1,5]=>4
[2,3,4,6,5,1]=>4
[2,3,5,1,4,6]=>3
[2,3,5,1,6,4]=>9
[2,3,5,4,1,6]=>3
[2,3,5,4,6,1]=>4
[2,3,5,6,1,4]=>9
[2,3,5,6,4,1]=>10
[2,3,6,1,4,5]=>6
[2,3,6,1,5,4]=>23
[2,3,6,4,1,5]=>7
[2,3,6,4,5,1]=>11
[2,3,6,5,1,4]=>26
[2,3,6,5,4,1]=>40
[2,4,1,3,5,6]=>2
[2,4,1,3,6,5]=>8
[2,4,1,5,3,6]=>5
[2,4,1,5,6,3]=>9
[2,4,1,6,3,5]=>16
[2,4,1,6,5,3]=>35
[2,4,3,1,5,6]=>2
[2,4,3,1,6,5]=>12
[2,4,3,5,1,6]=>3
[2,4,3,5,6,1]=>4
[2,4,3,6,1,5]=>16
[2,4,3,6,5,1]=>22
[2,4,5,1,3,6]=>5
[2,4,5,1,6,3]=>14
[2,4,5,3,1,6]=>6
[2,4,5,3,6,1]=>10
[2,4,5,6,1,3]=>14
[2,4,5,6,3,1]=>19
[2,4,6,1,3,5]=>16
[2,4,6,1,5,3]=>51
[2,4,6,3,1,5]=>24
[2,4,6,3,5,1]=>46
[2,4,6,5,1,3]=>56
[2,4,6,5,3,1]=>101
[2,5,1,3,4,6]=>3
[2,5,1,3,6,4]=>11
[2,5,1,4,3,6]=>9
[2,5,1,4,6,3]=>21
[2,5,1,6,3,4]=>21
[2,5,1,6,4,3]=>61
[2,5,3,1,4,6]=>4
[2,5,3,1,6,4]=>21
[2,5,3,4,1,6]=>7
[2,5,3,4,6,1]=>11
[2,5,3,6,1,4]=>30
[2,5,3,6,4,1]=>47
[2,5,4,1,3,6]=>11
[2,5,4,1,6,3]=>42
[2,5,4,3,1,6]=>20
[2,5,4,3,6,1]=>40
[2,5,4,6,1,3]=>56
[2,5,4,6,3,1]=>104
[2,5,6,1,3,4]=>21
[2,5,6,1,4,3]=>82
[2,5,6,3,1,4]=>40
[2,5,6,3,4,1]=>87
[2,5,6,4,1,3]=>117
[2,5,6,4,3,1]=>276
[2,6,1,3,4,5]=>4
[2,6,1,3,5,4]=>16
[2,6,1,4,3,5]=>16
[2,6,1,4,5,3]=>44
[2,6,1,5,3,4]=>38
[2,6,1,5,4,3]=>150
[2,6,3,1,4,5]=>7
[2,6,3,1,5,4]=>40
[2,6,3,4,1,5]=>14
[2,6,3,4,5,1]=>25
[2,6,3,5,1,4]=>66
[2,6,3,5,4,1]=>124
[2,6,4,1,3,5]=>24
[2,6,4,1,5,3]=>107
[2,6,4,3,1,5]=>54
[2,6,4,3,5,1]=>121
[2,6,4,5,1,3]=>163
[2,6,4,5,3,1]=>361
[2,6,5,1,3,4]=>49
[2,6,5,1,4,3]=>259
[2,6,5,3,1,4]=>130
[2,6,5,3,4,1]=>331
[2,6,5,4,1,3]=>481
[2,6,5,4,3,1]=>1356
[3,1,2,4,5,6]=>1
[3,1,2,4,6,5]=>3
[3,1,2,5,4,6]=>3
[3,1,2,5,6,4]=>6
[3,1,2,6,4,5]=>6
[3,1,2,6,5,4]=>17
[3,1,4,2,5,6]=>2
[3,1,4,2,6,5]=>8
[3,1,4,5,2,6]=>3
[3,1,4,5,6,2]=>4
[3,1,4,6,2,5]=>11
[3,1,4,6,5,2]=>16
[3,1,5,2,4,6]=>5
[3,1,5,2,6,4]=>16
[3,1,5,4,2,6]=>9
[3,1,5,4,6,2]=>16
[3,1,5,6,2,4]=>21
[3,1,5,6,4,2]=>38
[3,1,6,2,4,5]=>9
[3,1,6,2,5,4]=>35
[3,1,6,4,2,5]=>21
[3,1,6,4,5,2]=>44
[3,1,6,5,2,4]=>61
[3,1,6,5,4,2]=>150
[3,2,1,4,5,6]=>1
[3,2,1,4,6,5]=>6
[3,2,1,5,4,6]=>6
[3,2,1,5,6,4]=>17
[3,2,1,6,4,5]=>17
[3,2,1,6,5,4]=>66
[3,2,4,1,5,6]=>2
[3,2,4,1,6,5]=>12
[3,2,4,5,1,6]=>3
[3,2,4,5,6,1]=>4
[3,2,4,6,1,5]=>16
[3,2,4,6,5,1]=>22
[3,2,5,1,4,6]=>9
[3,2,5,1,6,4]=>35
[3,2,5,4,1,6]=>14
[3,2,5,4,6,1]=>22
[3,2,5,6,1,4]=>44
[3,2,5,6,4,1]=>67
[3,2,6,1,4,5]=>23
[3,2,6,1,5,4]=>112
[3,2,6,4,1,5]=>40
[3,2,6,4,5,1]=>73
[3,2,6,5,1,4]=>165
[3,2,6,5,4,1]=>318
[3,4,1,2,5,6]=>2
[3,4,1,2,6,5]=>10
[3,4,1,5,2,6]=>5
[3,4,1,5,6,2]=>9
[3,4,1,6,2,5]=>21
[3,4,1,6,5,2]=>44
[3,4,2,1,5,6]=>3
[3,4,2,1,6,5]=>22
[3,4,2,5,1,6]=>6
[3,4,2,5,6,1]=>10
[3,4,2,6,1,5]=>38
[3,4,2,6,5,1]=>67
[3,4,5,1,2,6]=>5
[3,4,5,1,6,2]=>14
[3,4,5,2,1,6]=>9
[3,4,5,2,6,1]=>19
[3,4,5,6,1,2]=>14
[3,4,5,6,2,1]=>28
[3,4,6,1,2,5]=>21
[3,4,6,1,5,2]=>65
[3,4,6,2,1,5]=>49
[3,4,6,2,5,1]=>116
[3,4,6,5,1,2]=>70
[3,4,6,5,2,1]=>183
[3,5,1,2,4,6]=>5
[3,5,1,2,6,4]=>21
[3,5,1,4,2,6]=>14
[3,5,1,4,6,2]=>30
[3,5,1,6,2,4]=>42
[3,5,1,6,4,2]=>105
[3,5,2,1,4,6]=>11
[3,5,2,1,6,4]=>61
[3,5,2,4,1,6]=>25
[3,5,2,4,6,1]=>47
[3,5,2,6,1,4]=>105
[3,5,2,6,4,1]=>208
[3,5,4,1,2,6]=>16
[3,5,4,1,6,2]=>56
[3,5,4,2,1,6]=>42
[3,5,4,2,6,1]=>104
[3,5,4,6,1,2]=>70
[3,5,4,6,2,1]=>186
[3,5,6,1,2,4]=>42
[3,5,6,1,4,2]=>147
[3,5,6,2,1,4]=>126
[3,5,6,2,4,1]=>334
[3,5,6,4,1,2]=>187
[3,5,6,4,2,1]=>618
[3,6,1,2,4,5]=>9
[3,6,1,2,5,4]=>44
[3,6,1,4,2,5]=>30
[3,6,1,4,5,2]=>74
[3,6,1,5,2,4]=>105
[3,6,1,5,4,2]=>320
[3,6,2,1,4,5]=>26
[3,6,2,1,5,4]=>165
[3,6,2,4,1,5]=>66
[3,6,2,4,5,1]=>139
[3,6,2,5,1,4]=>330
[3,6,2,5,4,1]=>761
[3,6,4,1,2,5]=>40
[3,6,4,1,5,2]=>163
[3,6,4,2,1,5]=>130
[3,6,4,2,5,1]=>363
[3,6,4,5,1,2]=>233
[3,6,4,5,2,1]=>739
[3,6,5,1,2,4]=>126
[3,6,5,1,4,2]=>555
[3,6,5,2,1,4]=>494
[3,6,5,2,4,1]=>1522
[3,6,5,4,1,2]=>887
[3,6,5,4,2,1]=>3473
[4,1,2,3,5,6]=>1
[4,1,2,3,6,5]=>4
[4,1,2,5,3,6]=>3
[4,1,2,5,6,3]=>6
[4,1,2,6,3,5]=>9
[4,1,2,6,5,3]=>23
[4,1,3,2,5,6]=>2
[4,1,3,2,6,5]=>12
[4,1,3,5,2,6]=>4
[4,1,3,5,6,2]=>7
[4,1,3,6,2,5]=>21
[4,1,3,6,5,2]=>40
[4,1,5,2,3,6]=>5
[4,1,5,2,6,3]=>16
[4,1,5,3,2,6]=>11
[4,1,5,3,6,2]=>24
[4,1,5,6,2,3]=>21
[4,1,5,6,3,2]=>49
[4,1,6,2,3,5]=>14
[4,1,6,2,5,3]=>51
[4,1,6,3,2,5]=>42
[4,1,6,3,5,2]=>107
[4,1,6,5,2,3]=>82
[4,1,6,5,3,2]=>259
[4,2,1,3,5,6]=>2
[4,2,1,3,6,5]=>12
[4,2,1,5,3,6]=>9
[4,2,1,5,6,3]=>23
[4,2,1,6,3,5]=>35
[4,2,1,6,5,3]=>112
[4,2,3,1,5,6]=>4
[4,2,3,1,6,5]=>28
[4,2,3,5,1,6]=>7
[4,2,3,5,6,1]=>11
[4,2,3,6,1,5]=>44
[4,2,3,6,5,1]=>73
[4,2,5,1,3,6]=>14
[4,2,5,1,6,3]=>51
[4,2,5,3,1,6]=>25
[4,2,5,3,6,1]=>46
[4,2,5,6,1,3]=>65
[4,2,5,6,3,1]=>116
[4,2,6,1,3,5]=>51
[4,2,6,1,5,3]=>214
[4,2,6,3,1,5]=>107
[4,2,6,3,5,1]=>226
[4,2,6,5,1,3]=>312
[4,2,6,5,3,1]=>693
[4,3,1,2,5,6]=>3
[4,3,1,2,6,5]=>22
[4,3,1,5,2,6]=>11
[4,3,1,5,6,2]=>26
[4,3,1,6,2,5]=>61
[4,3,1,6,5,2]=>165
[4,3,2,1,5,6]=>8
[4,3,2,1,6,5]=>72
[4,3,2,5,1,6]=>20
[4,3,2,5,6,1]=>40
[4,3,2,6,1,5]=>150
[4,3,2,6,5,1]=>318
[4,3,5,1,2,6]=>16
[4,3,5,1,6,2]=>56
[4,3,5,2,1,6]=>42
[4,3,5,2,6,1]=>101
[4,3,5,6,1,2]=>70
[4,3,5,6,2,1]=>183
[4,3,6,1,2,5]=>82
[4,3,6,1,5,2]=>312
[4,3,6,2,1,5]=>259
[4,3,6,2,5,1]=>693
[4,3,6,5,1,2]=>426
[4,3,6,5,2,1]=>1363
[4,5,1,2,3,6]=>5
[4,5,1,2,6,3]=>21
[4,5,1,3,2,6]=>16
[4,5,1,3,6,2]=>40
[4,5,1,6,2,3]=>42
[4,5,1,6,3,2]=>126
[4,5,2,1,3,6]=>16
[4,5,2,1,6,3]=>82
[4,5,2,3,1,6]=>41
[4,5,2,3,6,1]=>87
[4,5,2,6,1,3]=>147
[4,5,2,6,3,1]=>334
[4,5,3,1,2,6]=>28
[4,5,3,1,6,2]=>117
[4,5,3,2,1,6]=>99
[4,5,3,2,6,1]=>276
[4,5,3,6,1,2]=>187
[4,5,3,6,2,1]=>618
[4,5,6,1,2,3]=>42
[4,5,6,1,3,2]=>168
[4,5,6,2,1,3]=>168
[4,5,6,2,3,1]=>502
[4,5,6,3,1,2]=>315
[4,5,6,3,2,1]=>1306
[4,6,1,2,3,5]=>14
[4,6,1,2,5,3]=>65
[4,6,1,3,2,5]=>56
[4,6,1,3,5,2]=>163
[4,6,1,5,2,3]=>147
[4,6,1,5,3,2]=>555
[4,6,2,1,3,5]=>56
[4,6,2,1,5,3]=>312
[4,6,2,3,1,5]=>163
[4,6,2,3,5,1]=>389
[4,6,2,5,1,3]=>624
[4,6,2,5,3,1]=>1650
[4,6,3,1,2,5]=>117
[4,6,3,1,5,2]=>552
[4,6,3,2,1,5]=>481
[4,6,3,2,5,1]=>1490
[4,6,3,5,1,2]=>978
[4,6,3,5,2,1]=>3714
[4,6,5,1,2,3]=>168
[4,6,5,1,3,2]=>842
[4,6,5,2,1,3]=>830
[4,6,5,2,3,1]=>2866
[4,6,5,3,1,2]=>1902
[4,6,5,3,2,1]=>9048
[5,1,2,3,4,6]=>1
[5,1,2,3,6,4]=>4
[5,1,2,4,3,6]=>3
[5,1,2,4,6,3]=>7
[5,1,2,6,3,4]=>9
[5,1,2,6,4,3]=>26
[5,1,3,2,4,6]=>3
[5,1,3,2,6,4]=>16
[5,1,3,4,2,6]=>7
[5,1,3,4,6,2]=>14
[5,1,3,6,2,4]=>30
[5,1,3,6,4,2]=>66
[5,1,4,2,3,6]=>6
[5,1,4,2,6,3]=>24
[5,1,4,3,2,6]=>20
[5,1,4,3,6,2]=>54
[5,1,4,6,2,3]=>40
[5,1,4,6,3,2]=>130
[5,1,6,2,3,4]=>14
[5,1,6,2,4,3]=>56
[5,1,6,3,2,4]=>56
[5,1,6,3,4,2]=>163
[5,1,6,4,2,3]=>117
[5,1,6,4,3,2]=>481
[5,2,1,3,4,6]=>3
[5,2,1,3,6,4]=>16
[5,2,1,4,3,6]=>14
[5,2,1,4,6,3]=>40
[5,2,1,6,3,4]=>44
[5,2,1,6,4,3]=>165
[5,2,3,1,4,6]=>7
[5,2,3,1,6,4]=>44
[5,2,3,4,1,6]=>14
[5,2,3,4,6,1]=>25
[5,2,3,6,1,4]=>74
[5,2,3,6,4,1]=>139
[5,2,4,1,3,6]=>25
[5,2,4,1,6,3]=>107
[5,2,4,3,1,6]=>56
[5,2,4,3,6,1]=>121
[5,2,4,6,1,3]=>163
[5,2,4,6,3,1]=>363
[5,2,6,1,3,4]=>65
[5,2,6,1,4,3]=>312
[5,2,6,3,1,4]=>163
[5,2,6,3,4,1]=>389
[5,2,6,4,1,3]=>552
[5,2,6,4,3,1]=>1490
[5,3,1,2,4,6]=>6
[5,3,1,2,6,4]=>38
[5,3,1,4,2,6]=>25
[5,3,1,4,6,2]=>66
[5,3,1,6,2,4]=>105
[5,3,1,6,4,2]=>330
[5,3,2,1,4,6]=>20
[5,3,2,1,6,4]=>150
[5,3,2,4,1,6]=>56
[5,3,2,4,6,1]=>124
[5,3,2,6,1,4]=>320
[5,3,2,6,4,1]=>761
[5,3,4,1,2,6]=>41
[5,3,4,1,6,2]=>163
[5,3,4,2,1,6]=>133
[5,3,4,2,6,1]=>361
[5,3,4,6,1,2]=>233
[5,3,4,6,2,1]=>739
[5,3,6,1,2,4]=>147
[5,3,6,1,4,2]=>624
[5,3,6,2,1,4]=>555
[5,3,6,2,4,1]=>1650
[5,3,6,4,1,2]=>978
[5,3,6,4,2,1]=>3714
[5,4,1,2,3,6]=>9
[5,4,1,2,6,3]=>49
[5,4,1,3,2,6]=>42
[5,4,1,3,6,2]=>130
[5,4,1,6,2,3]=>126
[5,4,1,6,3,2]=>494
[5,4,2,1,3,6]=>42
[5,4,2,1,6,3]=>259
[5,4,2,3,1,6]=>133
[5,4,2,3,6,1]=>331
[5,4,2,6,1,3]=>555
[5,4,2,6,3,1]=>1522
[5,4,3,1,2,6]=>99
[5,4,3,1,6,2]=>481
[5,4,3,2,1,6]=>432
[5,4,3,2,6,1]=>1356
[5,4,3,6,1,2]=>887
[5,4,3,6,2,1]=>3473
[5,4,6,1,2,3]=>168
[5,4,6,1,3,2]=>830
[5,4,6,2,1,3]=>842
[5,4,6,2,3,1]=>2866
[5,4,6,3,1,2]=>1902
[5,4,6,3,2,1]=>9048
[5,6,1,2,3,4]=>14
[5,6,1,2,4,3]=>70
[5,6,1,3,2,4]=>70
[5,6,1,3,4,2]=>233
[5,6,1,4,2,3]=>187
[5,6,1,4,3,2]=>887
[5,6,2,1,3,4]=>70
[5,6,2,1,4,3]=>426
[5,6,2,3,1,4]=>233
[5,6,2,3,4,1]=>622
[5,6,2,4,1,3]=>978
[5,6,2,4,3,1]=>3020
[5,6,3,1,2,4]=>187
[5,6,3,1,4,2]=>978
[5,6,3,2,1,4]=>887
[5,6,3,2,4,1]=>3020
[5,6,3,4,1,2]=>1956
[5,6,3,4,2,1]=>8503
[5,6,4,1,2,3]=>315
[5,6,4,1,3,2]=>1902
[5,6,4,2,1,3]=>1902
[5,6,4,2,3,1]=>7425
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The number of connected components of long 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 long braid move if they are obtained from each other by a modification of the form $s_i s_{i+1} s_i \leftrightarrow s_{i+1} s_i s_{i+1}$ as a consecutive subword of a reduced word.
For example, the two reduced words $s_1s_3s_2s_3$ and $s_1s_2s_3s_2$ for
$$(124) = (12)(34)(23)(34) = (12)(23)(34)(23)$$
share an edge because they are obtained from each other by interchanging $s_3s_2s_3 \leftrightarrow s_3s_2s_3$.
This statistic counts the number connected components of such long 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 long braid move if they are obtained from each other by a modification of the form $s_i s_{i+1} s_i \leftrightarrow s_{i+1} s_i s_{i+1}$ as a consecutive subword of a reduced word.
For example, the two reduced words $s_1s_3s_2s_3$ and $s_1s_2s_3s_2$ for
$$(124) = (12)(34)(23)(34) = (12)(23)(34)(23)$$
share an edge because they are obtained from each other by interchanging $s_3s_2s_3 \leftrightarrow s_3s_2s_3$.
This statistic counts the number connected components of such long braid moves among all reduced words.
References
[1] Fishel, S., Milićević, E., Patrias, R., Tenner, B. E. Enumerations relating braid and commutation classes arXiv:1708.04372
Code
def long_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+3:] == w2[i+3:] and w1[i] != w2[i] and w1[i+2] != w2[i+2] and w1[i] == w1[i+2]for i in range(len(w1)-2) )
return Graph([V,is_edge])
def statistic(pi):
return len( long_braid_move_graph(pi).connected_components() )
Created
Jul 03, 2017 at 16:58 by Christian Stump
Updated
Mar 12, 2026 at 17:41 by Nupur Jain
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!