- FindStat

Identifier

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St001549: Permutations ⟶ ℤ

Values

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 9 + 4\ q + q^{2}$

$F_{5} = 21 + 15\ q + 5\ q^{2} + q^{3}$

Description

The number of restricted non-inversions between exceedances.
This is for a permutation

$\sigma$ of length

$n$ given by

$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \}.$

Map

Ringel

Description

The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.

Map

Corteel

Description

Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.

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.