- FindStat

Identifier

Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000660: Dyck paths ⟶ ℤ

Values

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

[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] => 1

[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] => 1

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

[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] => 1

[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] => 1

[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] => 1

[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] => 1

[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] => 1

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

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

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

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

[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] => 1

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

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

[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] => 1

[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] => 1

[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] => 1

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

[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] => 1

[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] => 1

[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] => 1

[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

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Description

The number of rises of length at least 3 of a Dyck path.
The number of Dyck paths without such rises are counted by the Motzkin numbers [1].

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.