- FindStat

Identifier

Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001137: Dyck paths ⟶ ℤ

Values

[1,0] => [1,0] => 0

[1,0,1,0] => [1,0,1,0] => 0

[1,1,0,0] => [1,1,0,0] => 0

[1,0,1,0,1,0] => [1,0,1,0,1,0] => 1

[1,0,1,1,0,0] => [1,1,0,1,0,0] => 0

[1,1,0,0,1,0] => [1,1,0,0,1,0] => 0

[1,1,0,1,0,0] => [1,0,1,1,0,0] => 0

[1,1,1,0,0,0] => [1,1,1,0,0,0] => 0

[1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => 0

[1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0] => 1

[1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 0

[1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,0] => 0

[1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 0

[1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => 1

[1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => 0

[1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 0

[1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,0] => 1

[1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,0,0] => 0

[1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => 0

[1,1,1,0,0,1,0,0] => [1,1,1,0,0,0,1,0] => 0

[1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0] => 0

[1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => 0

[1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => 0

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

[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,0,1,0] => 0

[1,0,1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => 0

[1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => 1

[1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,0,0,1,0,1,0] => 0

[1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => 0

[1,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0,1,0] => 0

[1,0,1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => 0

[1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => 0

[1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => 0

[1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 0

[1,0,1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => 0

[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 0

[1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 0

[1,1,0,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => 0

[1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 0

[1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 0

[1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,0,0,0] => 0

[1,1,0,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0] => 1

[1,1,0,1,0,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => 1

[1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => 1

[1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => 0

[1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0] => 1

[1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 0

[1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 0

[1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0] => 0

[1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => 0

[1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 1

[1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 0

[1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,0,1,0,0,1,0] => 0

[1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => 1

[1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 0

[1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => 0

[1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 0

[1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 1

[1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 0

[1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 0

[1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 0

[1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => 0

[1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 0

[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 0

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => 1

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,1,0,1,0,0] => 1

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

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => 0

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,1,0,0] => 0

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,1,0,0,1,0] => 0

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,1,0,1,0,0] => 0

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => 1

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => 1

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0,1,0] => 0

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,1,0,1,0,0,0] => 0

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => 1

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => 0

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,1,0,0] => 0

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0,1,0] => 0

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,1,0,0] => 0

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => 0

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,0,1,1,0,1,0,0,1,0,1,0] => 0

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,1,0,0] => 0

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,1,0,0,1,0] => 0

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,1,0,1,0,0] => 0

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => 1

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => 0

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0,1,0] => 0

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,1,0,1,0,0,0] => 0

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => 0

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => 1

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,1,0,0,0,1,1,0,0] => 0

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0,1,0] => 0

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0,1,0] => 0

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,1,0,0,0] => 0

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,1,0,0,1,0,0] => 0

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,1,0,1,0,0,0,1,0] => 0

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,1,0,0,0] => 0

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => 0

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Description

Number of simple modules that are 3-regular in the corresponding Nakayama algebra.

Map

Cori-Le Borgne involution

Description

The Cori-Le Borgne involution on Dyck paths.
Append an additional down step to the Dyck path and consider its (literal) reversal. The image of the involution is then the unique rotation of this word which is a Dyck word followed by an additional down step. Alternatively, it is the composite $\zeta\circ\mathrm{rev}\circ\zeta^{(-1)}$, where $\zeta$ is Mp00030zeta map.