- FindStat

Identifier

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ 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] => [1,1,0,0,1,0] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 9 + 5\ q$

$F_{5} = 21 + 21\ q$

$F_{6} = 51 + 78\ q + 3\ q^{2}$

$F_{7} = 127 + 274\ q + 28\ q^{2}$

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

Lalanne-Kreweras involution

Description

The Lalanne-Kreweras involution on Dyck paths.
Label the upsteps from left to right and record the labels on the first up step of each double rise. Do the same for the downsteps. Then form the Dyck path whose ascent lengths and descent lengths are the consecutives differences of the labels.