Identifier
Values
0 => [2] => [1,1,0,0] => [2,3,1] => 0
1 => [1,1] => [1,0,1,0] => [3,1,2] => 0
00 => [3] => [1,1,1,0,0,0] => [2,3,4,1] => 0
01 => [2,1] => [1,1,0,0,1,0] => [2,4,1,3] => 0
10 => [1,2] => [1,0,1,1,0,0] => [3,1,4,2] => 0
11 => [1,1,1] => [1,0,1,0,1,0] => [4,1,2,3] => 0
000 => [4] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 0
010 => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 0
011 => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 0
100 => [1,3] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
110 => [1,1,2] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 0
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 0
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 0
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 0
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 0
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 0
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 0
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 0
=> [1] => [1,0] => [2,1] => 0
search for individual values
searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
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
bounce path
Description
The bounce path determined by an integer composition.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.