- FindStat

Identifier

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000710: Permutations ⟶ ℤ

Values

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

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$F_{1} = 1$

$F_{2} = 1 + q$

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

$F_{4} = 1 + 6\ q + 6\ q^{2} + q^{3}$

Description

The number of big deficiencies of a permutation.
A big deficiency of a permutation

$\pi$ is an index

$i$ such that

$i - \pi(i) > 1$ .
This statistic is equidistributed with any of the numbers of big exceedences, big descents and big ascents.

Map

prime Dyck path

Description

Return the Dyck path obtained by adding an initial up and a final down step.

Map

Ringel

Description

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

Map

inverse promotion

Description

The inverse promotion of a Dyck path.
This is the bijection obtained by applying the inverse of Schützenberger's promotion to the corresponding two rowed standard Young tableau.