- FindStat

Identifier

Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000661: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 10 + 4\ q$

$F_{5} = 27 + 15\ q$

$F_{6} = 79 + 50\ q + 3\ q^{2}$

$F_{7} = 240 + 168\ q + 21\ q^{2}$

Description

The number of rises of length 3 of a Dyck path.

Map

inverse zeta map

Description

The inverse zeta map on Dyck paths.
See its inverse, the zeta map Mp00030zeta map, for the definition and details.