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Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000689: Dyck paths ⟶ ℤ

Values

[1] => [1,0,1,0] => [1,0,1,0] => [1,1,0,1,0,0] => 1
[2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 0
[1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 1
[3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 0
[2,1] => [1,0,1,0,1,0] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 1
[1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 1
[3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 0
[2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 1
[3,2] => [1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 0
[3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => 1
[2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => 1
[3,2,1] => [1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 2
[] => [] => [] => [1,0] => 0

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$F_{1} = q$

$F_{2} = 1 + q$

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

Description

The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid.
The correspondence between LNakayama algebras and Dyck paths is explained in St000684The global dimension of the LNakayama algebra associated to a Dyck path.. A module

$M$ is

$n$ -rigid, if

$\operatorname{Ext}^i(M,M)=0$ for

$1\leq i\leq n$ .
This statistic gives the maximal

$n$ such that the minimal generator-cogenerator module

$A \oplus D(A)$ of the LNakayama algebra

$A$ corresponding to a Dyck path is

$n$ -rigid.
An application is to check for maximal

$n$ -orthogonal objects in the module category in the sense of [2].

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.

Map

prime Dyck path

Description

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

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.