Identifier
Values
[1,0] => [2,1] => 0
[1,0,1,0] => [3,1,2] => 0
[1,1,0,0] => [2,3,1] => 0
[1,0,1,0,1,0] => [4,1,2,3] => 0
[1,0,1,1,0,0] => [3,1,4,2] => 0
[1,1,0,0,1,0] => [2,4,1,3] => 0
[1,1,0,1,0,0] => [4,3,1,2] => 2
[1,1,1,0,0,0] => [2,3,4,1] => 0
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[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] => 0
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 0
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 0
[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] => 5
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => 5
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 0
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 5
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 16
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 0
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 0
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 0
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 0
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 0
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 16
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 0
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 0
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 0
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => 11
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 0
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => 16
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => 9
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 0
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 0
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 11
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => 21
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => 14
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 0
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 9
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => 56
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 0
[] => [1] => 0
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
click to show known generating functions       
Description
The number 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 of such short braid moves among all reduced words.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.