- FindStat

Identifier

Mp00142: Dyck paths —promotion⟶ Dyck paths
St000026: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 5

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

[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 5

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

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

[1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 5

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

[1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,0] => 5

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

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

[1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => 5

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

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

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

[1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 5

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

[1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => 5

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

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

[1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => 5

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

[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => 5

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

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

[1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 5

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

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

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

[1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 5

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

[1,1,1,0,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 5

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

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

[1,1,1,0,1,0,0,0,1,0] => [1,1,1,1,0,1,0,0,0,0] => 5

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

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

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

[1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 5

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

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

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

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

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 6

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => 5

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => 6

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

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

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => 6

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0,1,0] => 5

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => 6

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

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

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => 6

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

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

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

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => 6

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,1,0,0,1,0] => 5

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => 6

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

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

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => 6

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,0,1,0] => 5

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => 6

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

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

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => 6

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

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

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

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 6

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => 5

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => 6

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

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

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => 6

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

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

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

>>> Load all 625 entries. <<<

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$F_{1} = q$

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

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

$F_{4} = 5\ q + 2\ q^{2} + 2\ q^{3} + 5\ q^{4}$

$F_{5} = 14\ q + 5\ q^{2} + 4\ q^{3} + 5\ q^{4} + 14\ q^{5}$

$F_{6} = 42\ q + 14\ q^{2} + 10\ q^{3} + 10\ q^{4} + 14\ q^{5} + 42\ q^{6}$

$F_{7} = 132\ q + 42\ q^{2} + 28\ q^{3} + 25\ q^{4} + 28\ q^{5} + 42\ q^{6} + 132\ q^{7}$

Description

The position of the first return of a Dyck path.

Map

promotion

Description

The promotion of the two-row standard Young tableau of a Dyck path.
Dyck paths of semilength

$n$ are in bijection with standard Young tableaux of shape

$(n^2)$ , see Mp00033to two-row standard tableau.
This map is the bijection on such standard Young tableaux given by Schützenberger's promotion. For definitions and details, see [1] and the references therein.