- FindStat

Identifier

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000007: Permutations ⟶ ℤ

Values

[1] => [1,0] => [1,0] => [2,1] => 2
[2] => [1,0,1,0] => [1,1,0,0] => [2,3,1] => 2
[1,1] => [1,1,0,0] => [1,0,1,0] => [3,1,2] => 2
[3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => [2,3,4,1] => 2
[2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => [2,4,1,3] => 2
[1,1,1] => [1,1,0,1,0,0] => [1,1,0,1,0,0] => [4,3,1,2] => 3
[4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 2
[3,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 2
[2,2] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => [4,1,2,3] => 2
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 3
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 3
[5] => [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] => 2
[4,1] => [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] => 2
[3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 2
[3,1,1] => [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] => 3
[2,2,1] => [1,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 3
[2,1,1,1] => [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] => 3
[1,1,1,1,1] => [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] => 3
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => 2
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => 2
[4,2] => [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] => 2
[3,3] => [1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => 3
[3,2,1] => [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] => 3
[2,2,2] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 2
[2,2,1,1] => [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] => 3
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => 2
[6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [2,3,4,5,6,8,1,7] => 2
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,4,7,1,5,6] => 2
[4,3] => [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] => 3
[3,3,1] => [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] => 3
[3,2,2] => [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] => 2
[2,2,2,1] => [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] => 3
[8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => 2
[7,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [2,3,4,5,6,7,9,1,8] => 2
[4,4] => [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] => 4
[3,3,2] => [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] => 2
[2,2,2,2] => [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] => 2
[9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,1] => 2
[3,3,3] => [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] => 2
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => [2,6,7,1,3,4,5] => 2
[10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => 2
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => [7,6,4,5,1,2,3] => 4
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => 2
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => [7,4,1,2,3,5,6] => 2
[6,6] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [8,7,4,5,6,1,2,3] => 4
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => 3
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => 2
[4,3,3,3] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [2,8,1,3,4,5,6,7] => 2
[4,4,3,3] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0] => [8,4,1,2,3,5,6,7] => 2
[5,5,5] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [6,7,8,1,2,3,4,5] => 2
[3,3,3,3,3] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [8,7,1,2,3,4,5,6] => 3
[4,4,4,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,1,2,3,4,5,6,7] => 2
[5,4,4,4] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,9,1,3,4,5,6,7,8] => 2
[] => [] => [] => [1] => 1
[3,3,3,3,3,3] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => [9,7,8,1,2,3,4,5,6] => 3
[3,3,3,3,3,3,3] => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0] => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0] => [7,8,9,10,1,2,3,4,5,6] => 2
[4,4,4,4,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [9,1,2,3,4,5,6,7,8] => 2
[4,4,4,4,4,4] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [10,9,1,2,3,4,5,6,7,8] => 3
[5,5,5,5] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [8,9,1,2,3,4,5,6,7] => 2
[6,6,6,6] => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => [10,9,8,1,2,3,4,5,6,7] => 4
[5,5,5,5,5,5] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [11,1,2,3,4,5,6,7,8,9,10] => 2
[5,5,5,5,5] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [10,1,2,3,4,5,6,7,8,9] => 2
[5,4,4,4,4] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,10,1,3,4,5,6,7,8,9] => 2

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$F_{1} = q^{2}$

$F_{2} = 2\ q^{2}$

$F_{3} = 2\ q^{2} + q^{3}$

$F_{4} = 3\ q^{2} + 2\ q^{3}$

$F_{5} = 3\ q^{2} + 4\ q^{3}$

Description

The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern

$([1], {(1,1)})$ , i.e., the upper right quadrant is shaded, see [1].

Map

parallelogram polyomino

Description

Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.

Map

Ringel

Description

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

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.