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
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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
Jul 03, 2017 at 16:58 by Christian Stump
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