- FindStat

Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00181: Skew partitions —row lengths⟶ Integer compositions
St001235: Integer compositions ⟶ ℤ

Values

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

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

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

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

Description

The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".

Map

row lengths

Description

The sequence of row lengths from top to bottom.

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

$\gamma(D)$ .

Map

to Dyck path

Description

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