- FindStat

Identifier

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00277: Permutations —catalanization⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000996: Permutations ⟶ ℤ

Values

[1,0] => [2,1] => [2,1] => [1,2] => 0

[1,0,1,0] => [3,1,2] => [2,3,1] => [1,2,3] => 0

[1,1,0,0] => [2,3,1] => [2,3,1] => [1,2,3] => 0

[1,0,1,0,1,0] => [4,1,2,3] => [2,3,4,1] => [1,2,3,4] => 0

[1,0,1,1,0,0] => [3,1,4,2] => [2,3,4,1] => [1,2,3,4] => 0

[1,1,0,0,1,0] => [2,4,1,3] => [4,3,1,2] => [4,2,1,3] => 1

[1,1,0,1,0,0] => [4,3,1,2] => [3,4,2,1] => [3,1,2,4] => 1

[1,1,1,0,0,0] => [2,3,4,1] => [2,3,4,1] => [1,2,3,4] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 3 + 2\ q$

$F_{4} = 4 + 10\ q$

$F_{5} = 5 + 33\ q + 4\ q^{2}$

Description

The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.

Map

Inverse Kreweras complement

Description

Sends the permutation

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

$c\pi^{-1}$ where

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

Map

Ringel

Description

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

Map

catalanization

Description

The catalanization of a permutation.
For a permutation

$\sigma$ , this is the product of the reflections corresponding to the inversions of

$\sigma$ in lex-order.
A permutation is

$231$ -avoiding if and only if it is a fixpoint of this map. Also, for every permutation there exists an index

$k$ such that the

$k$ -fold application of this map is

$231$ -avoiding.