- FindStat

Identifier

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000356: Permutations ⟶ ℤ

Values

[1] => [1,0] => [2,1] => [1,2] => 0
[1,1] => [1,0,1,0] => [3,1,2] => [3,1,2] => 0
[2] => [1,1,0,0] => [2,3,1] => [1,2,3] => 0
[1,1,1] => [1,0,1,0,1,0] => [4,1,2,3] => [3,4,1,2] => 0
[1,2] => [1,0,1,1,0,0] => [3,1,4,2] => [3,1,2,4] => 0
[2,1] => [1,1,0,0,1,0] => [2,4,1,3] => [4,2,1,3] => 0
[3] => [1,1,1,0,0,0] => [2,3,4,1] => [1,2,3,4] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [3,4,5,1,2] => 0
[1,1,2] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [3,4,1,2,5] => 0
[1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [3,5,2,1,4] => 1
[1,3] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [3,1,2,4,5] => 0
[2,1,1] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [4,2,5,1,3] => 1
[2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [4,2,1,3,5] => 0
[3,1] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [5,2,3,1,4] => 0
[4] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [3,4,5,6,1,2] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [3,4,5,1,2,6] => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [3,4,6,2,1,5] => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [3,4,1,2,5,6] => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [3,5,2,6,1,4] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [3,5,2,1,4,6] => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [3,6,2,4,1,5] => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [3,1,2,4,5,6] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [4,2,5,6,1,3] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [4,2,5,1,3,6] => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [4,2,6,3,1,5] => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [4,2,1,3,5,6] => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [5,2,3,6,1,4] => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [5,2,3,1,4,6] => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [6,2,3,4,1,5] => 0
[5] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [3,4,5,6,7,1,2] => 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [4,1,2,5,6,7,3] => [3,4,1,2,5,6,7] => 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [3,1,4,5,6,7,2] => [3,1,2,4,5,6,7] => 0
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [2,5,1,3,6,7,4] => [4,2,5,1,3,6,7] => 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => [7,2,3,4,5,1,6] => 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => 0
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,1,2,3,4,5,6,7] => [3,4,5,6,7,8,1,2] => 0
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [4,1,2,5,6,7,8,3] => [3,4,1,2,5,6,7,8] => 0
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [2,5,1,3,6,7,8,4] => [4,2,5,1,3,6,7,8] => 1
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => [1,2,3,4,5,6,7,8] => 0
[1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [4,1,2,5,6,7,8,9,3] => [3,4,1,2,5,6,7,8,9] => 0
[2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0] => [2,5,1,3,6,7,8,9,4] => [4,2,5,1,3,6,7,8,9] => 1
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => [1,2,3,4,5,6,7,8,9] => 0
[1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [4,1,2,5,6,7,8,9,10,3] => [3,4,1,2,5,6,7,8,9,10] => 0
[9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,1] => [1,2,3,4,5,6,7,8,9,10] => 0

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4$

$F_{4} = 6 + 2\ q$

$F_{5} = 8 + 5\ q + 3\ q^{2}$

Description

The number of occurrences of the pattern 13-2.
See Permutations/#Pattern-avoiding_permutations for the definition of the pattern

$13\!\!-\!\!2$ .

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

Kreweras complement

Description

Sends the permutation

$\pi \in \mathfrak{S}_n$ to the permutation

$\pi^{-1}c$ where

$c = (1,\ldots,n)$ is the long cycle.