Identifier
Values
[1] => [1,0] => [1,0] => [2,1] => 1
[1,1] => [1,0,1,0] => [1,1,0,0] => [2,3,1] => 2
[2] => [1,1,0,0] => [1,0,1,0] => [3,1,2] => 2
[1,1,1] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => [4,3,1,2] => 3
[1,2] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => [2,4,1,3] => 3
[2,1] => [1,1,0,0,1,0] => [1,1,1,0,0,0] => [2,3,4,1] => 3
[3] => [1,1,1,0,0,0] => [1,0,1,1,0,0] => [3,1,4,2] => 3
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => 3
[1,1,2] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 4
[1,2,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => 4
[1,3] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 4
[2,1,1] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 4
[2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 4
[3,1] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 4
[4] => [1,1,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 4
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => 4
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => 4
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => 5
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => 5
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => 5
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => 5
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 5
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => 4
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => 5
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 5
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 5
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 5
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 5
[5] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 5
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 absolute length of a permutation.
The absolute length of a permutation $\sigma$ of length $n$ is the shortest $k$ such that $\sigma = t_1 \dots t_k$ for transpositions $t_i$. Also, this is equal to $n$ minus the number of cycles of $\sigma$.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
promotion
Description
The promotion of the two-row standard Young tableau of a Dyck path.
Dyck paths of semilength $n$ are in bijection with standard Young tableaux of shape $(n^2)$, see Mp00033to two-row standard tableau.
This map is the bijection on such standard Young tableaux given by Schützenberger's promotion. For definitions and details, see [1] and the references therein.