- FindStat

Identifier

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St001269: Permutations ⟶ ℤ

Values

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

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 3\ q + q^{2}$

$F_{4} = 1 + 6\ q + 7\ q^{2}$

$F_{5} = 1 + 10\ q + 26\ q^{2} + 5\ q^{3}$

Description

The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation.

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.