- FindStat

Identifier

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤ

Values

{{1}} => [1] => [1,0] => [[1],[]] => 1
{{1,2}} => [2,1] => [1,1,0,0] => [[2],[]] => 1
{{1},{2}} => [1,2] => [1,0,1,0] => [[1,1],[]] => 1
{{1,2,3}} => [2,3,1] => [1,1,0,1,0,0] => [[3],[]] => 1
{{1,2},{3}} => [2,1,3] => [1,1,0,0,1,0] => [[2,2],[1]] => 1
{{1,3},{2}} => [3,2,1] => [1,1,1,0,0,0] => [[2,2],[]] => 1
{{1},{2,3}} => [1,3,2] => [1,0,1,1,0,0] => [[2,1],[]] => 1
{{1},{2},{3}} => [1,2,3] => [1,0,1,0,1,0] => [[1,1,1],[]] => 1
{{1,2,3,4}} => [2,3,4,1] => [1,1,0,1,0,1,0,0] => [[4],[]] => 1
{{1,2,3},{4}} => [2,3,1,4] => [1,1,0,1,0,0,1,0] => [[3,3],[2]] => 1
{{1,2,4},{3}} => [2,4,3,1] => [1,1,0,1,1,0,0,0] => [[3,3],[1]] => 1
{{1,2},{3,4}} => [2,1,4,3] => [1,1,0,0,1,1,0,0] => [[3,2],[1]] => 1
{{1,2},{3},{4}} => [2,1,3,4] => [1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]] => 1
{{1,3,4},{2}} => [3,2,4,1] => [1,1,1,0,0,1,0,0] => [[3,2],[]] => 1
{{1,3},{2},{4}} => [3,2,1,4] => [1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => 1
{{1},{2,3,4}} => [1,3,4,2] => [1,0,1,1,0,1,0,0] => [[3,1],[]] => 1
{{1},{2,3},{4}} => [1,3,2,4] => [1,0,1,1,0,0,1,0] => [[2,2,1],[1]] => 1
{{1},{2,4},{3}} => [1,4,3,2] => [1,0,1,1,1,0,0,0] => [[2,2,1],[]] => 1
{{1},{2},{3,4}} => [1,2,4,3] => [1,0,1,0,1,1,0,0] => [[2,1,1],[]] => 1
{{1},{2},{3},{4}} => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 1
{{1,2,3,4,5}} => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => [[5],[]] => 1
{{1,2,3,4},{5}} => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => 1
{{1,2,3},{4,5}} => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0] => [[4,3],[2]] => 1
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => [[3,3,3],[2,2]] => 1
{{1,2},{3,4,5}} => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0] => [[4,2],[1]] => 1
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0] => [[3,3,2],[2,1]] => 1
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0] => [[3,2,2],[1,1]] => 1
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => 1
{{1},{2,3,4,5}} => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => [[4,1],[]] => 1
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => [[3,3,1],[2]] => 1
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => [[3,2,1],[1]] => 1
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => 1
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [[3,1,1],[]] => 1
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => 1
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => 1
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1

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$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 5\ q$

Description

The number of connected components of a skew partition.

Map

to permutation

Description

Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.

Map

left-to-right-maxima to Dyck path

Description

The left-to-right maxima of a permutation as a Dyck path.
Let

$(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are

$c_1, c_1+c_2, \dots, c_1+\dots+c_k$ .
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.

Map

skew partition

Description

The parallelogram polyomino corresponding to a Dyck path, interpreted as a skew partition.
Let

$D$ be a Dyck path of semilength

$n$ . The parallelogram polyomino

$\gamma(D)$ is defined as follows: let

$\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to

$D$ . Then, the upper path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with even indices, and the lower path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with odd indices.
This map returns the skew partition definded by the diagram of

$\gamma(D)$ .