- FindStat

Identifier

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000664: 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] => [1,0,1,1,0,0] => [3,1,4,2] => 0
[1,1,0,0] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => [4,1,2,3] => 0
[1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 0
[1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 0
[1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 1
[1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 0
[1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => 0
[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] => 0
[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] => 1
[1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => 0
[1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 0
[1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 0
[1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 0
[1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 1
[1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => 1
[1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 0
[1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 1
[1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 1
[1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 0
[] => [1,0] => [1,0] => [2,1] => 0

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

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

Description

The number of right ropes of a permutation.
Let

$\pi$ be a permutation of length

$n$ . A raft of

$\pi$ is a non-empty maximal sequence of consecutive small ascents, St000441The number of successions of a permutation., and a right rope is a large ascent after a raft of

$\pi$ .
See Definition 3.10 and Example 3.11 in [1].

Map

prime Dyck path

Description

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

Map

Elizalde-Deutsch bijection

Description

The Elizalde-Deutsch bijection on Dyck paths.
.Let

$n$ be the length of the Dyck path. Consider the steps

$1,n,2,n-1,\dots$ of

$D$ . When considering the

$i$ -th step its corresponding matching step has not yet been read, let the

$i$ -th step of the image of

$D$ be an up step, otherwise let it be a down step.

Map

Ringel

Description

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