- FindStat

Identifier

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000365: Permutations ⟶ ℤ

Values

[1,0] => [1,0] => [2,1] => [2,1] => 0
[1,0,1,0] => [1,1,0,0] => [2,3,1] => [3,1,2] => 0
[1,1,0,0] => [1,0,1,0] => [3,1,2] => [3,2,1] => 0
[1,0,1,0,1,0] => [1,1,1,0,0,0] => [2,3,4,1] => [4,1,2,3] => 1
[1,0,1,1,0,0] => [1,1,0,0,1,0] => [2,4,1,3] => [4,3,1,2] => 0
[1,1,0,0,1,0] => [1,0,1,1,0,0] => [3,1,4,2] => [4,2,1,3] => 0
[1,1,0,1,0,0] => [1,1,0,1,0,0] => [4,3,1,2] => [4,2,3,1] => 0
[1,1,1,0,0,0] => [1,0,1,0,1,0] => [4,1,2,3] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => 2
[1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [5,4,1,2,3] => 1
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [5,3,1,2,4] => 1
[1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [5,3,4,1,2] => 0
[1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [5,4,3,1,2] => 0
[1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [5,2,1,3,4] => 1
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [5,4,2,1,3] => 0
[1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [5,2,3,1,4] => 0
[1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [5,2,3,4,1] => 1
[1,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [5,4,2,3,1] => 0
[1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [5,3,2,1,4] => 0
[1,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [5,3,4,2,1] => 0
[1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [4,2,5,3,1] => 0
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 3
[1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [6,5,1,2,3,4] => 2
[1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [6,4,1,2,3,5] => 2
[1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [6,4,5,1,2,3] => 1
[1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [6,5,4,1,2,3] => 1
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [6,3,1,2,4,5] => 2
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [6,5,3,1,2,4] => 1
[1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [6,3,4,1,2,5] => 1
[1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [6,3,4,5,1,2] => 1
[1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [6,5,3,4,1,2] => 0
[1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [6,4,3,1,2,5] => 1
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [6,4,5,3,1,2] => 0
[1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [5,3,6,4,1,2] => 0
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [6,5,4,3,1,2] => 0
[1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [6,2,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [6,5,2,1,3,4] => 1
[1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [6,4,2,1,3,5] => 1
[1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [6,4,5,2,1,3] => 0
[1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [6,5,4,2,1,3] => 0
[1,1,0,1,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [6,2,3,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [6,5,2,3,1,4] => 0
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [6,2,3,4,1,5] => 1
[1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [6,2,3,4,5,1] => 2
[1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [6,5,2,3,4,1] => 1
[1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [6,4,2,3,1,5] => 0
[1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [6,4,5,2,3,1] => 0
[1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [5,2,3,6,4,1] => 1
[1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [6,5,4,2,3,1] => 0
[1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [6,3,2,1,4,5] => 1
[1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [6,5,3,2,1,4] => 0
[1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [6,3,4,2,1,5] => 0
[1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [6,3,4,5,2,1] => 1
[1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [6,5,3,4,2,1] => 0
[1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [4,2,6,3,1,5] => 0
[1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [5,2,4,6,3,1] => 1
[1,1,1,0,1,0,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [5,2,6,3,4,1] => 0
[1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [4,2,6,5,3,1] => 0
[1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [6,4,3,2,1,5] => 0
[1,1,1,1,0,0,0,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [6,4,5,3,2,1] => 0
[1,1,1,1,0,0,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [5,3,6,4,2,1] => 0
[1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [5,3,1,6,4,2] => 0
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [6,5,4,3,2,1] => 0
[] => [] => [1] => [1] => 0

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 9 + 4\ q + q^{2}$

$F_{5} = 21 + 15\ q + 5\ q^{2} + q^{3}$

Description

The number of double ascents of a permutation.
A double ascent of a permutation

$\pi$ is a position

$i$ such that

$\pi(i) < \pi(i+1) < \pi(i+2)$ .

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.

Map

inverse first fundamental transformation

Description

Let

$\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps

$\sigma$ to the permutation

$[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.

Map

Ringel

Description

The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.