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Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000541: Permutations ⟶ ℤ (values match St000542The number of left-to-right-minima of a permutation.)

Values

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

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

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

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

Description

The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.
For a permutation

$\pi$ of length

$n$ , this is the number of indices

$2 \leq j \leq n$ such that for all

$1 \leq i < j$ , the pair

$(i,j)$ is an inversion of

$\pi$ .

Map

prime Dyck path

Description

Return the Dyck path obtained by adding an initial up and a final down step.

Map

Ringel

Description

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

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.