Identifier
Values
[1,0] => [2,1] => 1
[1,0,1,0] => [3,1,2] => 1
[1,1,0,0] => [2,3,1] => 1
[1,0,1,0,1,0] => [4,1,2,3] => 1
[1,0,1,1,0,0] => [3,1,4,2] => 1
[1,1,0,0,1,0] => [2,4,1,3] => 1
[1,1,0,1,0,0] => [4,3,1,2] => 3
[1,1,1,0,0,0] => [2,3,4,1] => 1
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 1
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 3
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 1
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 1
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 3
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => 4
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => 3
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 1
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 3
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 6
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 1
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 1
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 1
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 1
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 3
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 1
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 1
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 1
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => 3
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 1
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => 3
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => 3
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 1
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 1
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 3
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => 4
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => 3
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 1
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 3
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => 6
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 1
[] => [1] => 1
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Description
The number of connected components 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 connected components of such short braid moves among all reduced words.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.