- FindStat

Identifier

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000920: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 2

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The logarithmic height of a Dyck path.
This is the floor of the binary logarithm of the usual height increased by one:
$$ \lfloor\log_2(1+height(D))\rfloor $$

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.