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
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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
Aug 24, 2017 at 23:36 by Martin Rubey
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