- FindStat

Identifier

Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000022: Permutations ⟶ ℤ

Values

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

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$F_{1} = 1$

$F_{2} = 2$

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

$F_{4} = 9 + 5\ q$

$F_{5} = 27 + 15\ q$

Description

The number of fixed points of a permutation.

Map

Simion-Schmidt map

Description

The Simion-Schmidt map sends any permutation to a

$123$ -avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between

$132$ -avoiding permutations and

$123$ -avoiding permutations, see [1, Proposition 19].

Map

Ringel

Description

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

Map

reflect parallelogram polyomino

Description

Reflect the corresponding parallelogram polyomino, such that the first column becomes the first row.
Let

$D$ be a Dyck path of semilength

$n$ . The parallelogram polyomino

$\gamma(D)$ is defined as follows: let

$\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to

$D$ . Then, the upper path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with even indices, and the lower path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with odd indices.