- FindStat

Identifier

Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000445: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

[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] => 4

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

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

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

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

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

[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] => 2

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

[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] => 1

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

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

[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] => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 3

[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] => 3

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

[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] => 1

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

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

[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] => 2

[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] => 2

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

[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] => 2

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

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

[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] => 1

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

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

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

[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] => 6

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 4

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => 4

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,1,0,1,0,0] => 4

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

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => 4

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

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,1,0,0,1,0] => 4

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,1,0,1,0,0] => 4

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

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

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

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

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

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => 4

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

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

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

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

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,0,1,1,0,1,0,0,1,0,1,0] => 4

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

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,1,0,0,1,0] => 4

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,1,0,1,0,0] => 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

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

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

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

$F_{5} = 6 + 15\ q + 10\ q^{2} + 10\ q^{3} + q^{5}$

$F_{6} = 15 + 36\ q + 45\ q^{2} + 20\ q^{3} + 15\ q^{4} + q^{6}$

$F_{7} = 36 + 105\ q + 126\ q^{2} + 105\ q^{3} + 35\ q^{4} + 21\ q^{5} + q^{7}$

Description

The number of rises of length 1 of a Dyck path.

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.