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
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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
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.
Map
bounce path
Description
The bounce path determined by an integer composition.