Identifier
Values
[1] => [1,0] => [1,0] => [2,1] => 0
[2] => [1,0,1,0] => [1,0,1,0] => [3,1,2] => 0
[1,1] => [1,1,0,0] => [1,1,0,0] => [2,3,1] => 0
[3] => [1,0,1,0,1,0] => [1,0,1,0,1,0] => [4,1,2,3] => 0
[2,1] => [1,0,1,1,0,0] => [1,0,1,1,0,0] => [3,1,4,2] => 0
[1,1,1] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => [3,1,4,2] => 0
[4] => [1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[3,1] => [1,0,1,0,1,1,0,0] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[2,2] => [1,1,1,0,0,0] => [1,1,1,0,0,0] => [2,3,4,1] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 0
[3,2] => [1,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 0
[2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 0
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 0
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 0
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 0
[3,3] => [1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 0
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 0
[2,2,2] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 0
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 0
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 0
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 0
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 0
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 0
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 0
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 0
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 0
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [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
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.
Map
bounce path
Description
Sends a Dyck path $D$ of length $2n$ to its bounce path.
This path is formed by starting at the endpoint $(n,n)$ of $D$ and traveling west until encountering the first vertical step of $D$, then south until hitting the diagonal, then west again until hitting another vertical step of $D$, etc. until the point $(0,0)$ is reached.
This map is the first part of the zeta map Mp00030zeta map.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.