- FindStat

Identifier

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000216: Permutations ⟶ ℤ

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

$F_{1} = q$

$F_{2} = 2\ q^{2}$

$F_{3} = 4\ q^{3}$

$F_{4} = q^{3} + 7\ q^{4}$

$F_{5} = 4\ q^{4} + 12\ q^{5}$

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

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.

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.