- FindStat

Identifier

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001232: Dyck paths ⟶ ℤ

Values

[1,0] => [1,1,0,0] => [1,1,0,0] => [1,1,0,0] => 0
[1,1,0,0] => [1,1,1,0,0,0] => [1,1,1,0,0,0] => [1,1,1,0,0,0] => 0
[1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 3
[1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => 0
[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 4
[1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => 4
[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] => [1,1,0,1,1,0,0,0,1,0] => 5
[1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 5
[1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 0
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => 5
[1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,1,0,0] => 6
[1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7
[1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => [1,0,1,1,1,0,0,1,1,0,0,0] => 7
[1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => 7
[1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => [1,0,1,1,1,1,0,0,1,0,0,0] => [1,0,1,1,1,1,0,0,1,0,0,0] => 7
[1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,1,0,0,0,1,0,0] => 6
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
[1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,1,1,1,0,0,0,1,0,0,0] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 6
[1,0,1,1,1,1,0,0,1,0,0,0] => [1,1,0,1,1,1,1,0,0,1,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 6
[1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,1,1,1,0,1,0,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 6
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6
[1,1,0,0,1,1,1,0,0,1,0,0] => [1,1,1,0,0,1,1,1,0,0,1,0,0,0] => [1,1,1,1,0,0,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => 5
[1,1,0,0,1,1,1,1,0,0,0,0] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0,1,1,0,0] => [1,1,1,0,0,1,1,1,0,0,0,1,0,0] => 8
[1,1,0,1,1,1,0,0,1,0,0,0] => [1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,1,0,0,0] => [1,1,0,1,1,0,0,0,1,1,1,0,0,0] => 7
[1,1,0,1,1,1,0,1,0,0,0,0] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0,1,1,0,0] => 8
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0] => 9
[1,1,1,0,1,1,0,0,0,1,0,0] => [1,1,1,1,0,1,1,0,0,0,1,0,0,0] => [1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [1,0,1,1,1,0,0,1,1,1,0,0,0,0] => 9
[1,1,1,0,1,1,0,1,0,0,0,0] => [1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [1,1,1,0,1,1,1,0,0,0,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0,1,1,0,0] => 8
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0] => 10
[1,1,1,1,0,1,0,0,0,1,0,0] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0] => [1,1,0,1,1,1,1,0,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0] => 10
[1,1,1,1,0,1,0,0,1,0,0,0] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [1,1,0,1,1,1,1,0,0,0,1,0,0,0] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0] => 8
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0] => 9
[1,1,1,1,1,0,0,0,0,1,0,0] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9
[1,1,1,1,1,0,0,0,1,0,0,0] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [1,0,1,1,1,1,1,0,0,0,1,0,0,0] => [1,0,1,1,1,1,1,0,0,0,1,0,0,0] => 8
[1,1,1,1,1,0,0,1,0,0,0,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => 7
[1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 0
[] => [1,0] => [1,0] => [1,0] => 0

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$F_{1} = 1$

Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Map

Barnabei-Castronuovo involution

Description

The Barnabei-Castronuovo Schützenberger involution on Dyck paths.
The image of a Dyck path is obtained by reversing the canonical decompositions of the two halves of the Dyck path. More precisely, let

$D_1, 1, D_2, 1, \dots$ be the canonical decomposition of the first half, then the canonical decomposition of the first half of the image is

$\dots, 1, D_2, 1, D_1$ .

Map

prime Dyck path

Description

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

Map

switch returns and last double rise

Description

An alternative to the Adin-Bagno-Roichman transformation of a Dyck path.
This is a bijection preserving the number of up steps before each peak and exchanging the number of components with the position of the last double rise.