- FindStat

Identifier

Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00181: Skew partitions —row lengths⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000708: Integer partitions ⟶ ℤ

Values

[1,0,1,0] => [[1,1],[]] => [1,1] => [1,1] => 1

[1,1,0,0] => [[2],[]] => [2] => [2] => 2

[1,0,1,0,1,0] => [[1,1,1],[]] => [1,1,1] => [1,1,1] => 1

[1,0,1,1,0,0] => [[2,1],[]] => [2,1] => [2,1] => 2

[1,1,0,0,1,0] => [[2,2],[1]] => [1,2] => [2,1] => 2

[1,1,0,1,0,0] => [[3],[]] => [3] => [3] => 3

[1,1,1,0,0,0] => [[2,2],[]] => [2,2] => [2,2] => 4

[1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => [1,1,1,1] => [1,1,1,1] => 1

[1,0,1,0,1,1,0,0] => [[2,1,1],[]] => [2,1,1] => [2,1,1] => 2

[1,0,1,1,0,0,1,0] => [[2,2,1],[1]] => [1,2,1] => [2,1,1] => 2

[1,0,1,1,0,1,0,0] => [[3,1],[]] => [3,1] => [3,1] => 3

[1,0,1,1,1,0,0,0] => [[2,2,1],[]] => [2,2,1] => [2,2,1] => 4

[1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]] => [1,1,2] => [2,1,1] => 2

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

[1,1,0,1,0,0,1,0] => [[3,3],[2]] => [1,3] => [3,1] => 3

[1,1,0,1,0,1,0,0] => [[4],[]] => [4] => [4] => 4

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

[1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => [1,2,2] => [2,2,1] => 4

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

[1,1,1,0,1,0,0,0] => [[2,2,2],[]] => [2,2,2] => [2,2,2] => 8

[1,1,1,1,0,0,0,0] => [[3,3],[]] => [3,3] => [3,3] => 9

[1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => [1,1,1,1,1] => [1,1,1,1,1] => 1

[1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => [2,1,1,1] => [2,1,1,1] => 2

[1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => [1,2,1,1] => [2,1,1,1] => 2

[1,0,1,0,1,1,0,1,0,0] => [[3,1,1],[]] => [3,1,1] => [3,1,1] => 3

[1,0,1,0,1,1,1,0,0,0] => [[2,2,1,1],[]] => [2,2,1,1] => [2,2,1,1] => 4

[1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => [1,1,2,1] => [2,1,1,1] => 2

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

[1,0,1,1,0,1,0,0,1,0] => [[3,3,1],[2]] => [1,3,1] => [3,1,1] => 3

[1,0,1,1,0,1,0,1,0,0] => [[4,1],[]] => [4,1] => [4,1] => 4

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

[1,0,1,1,1,0,0,0,1,0] => [[2,2,2,1],[1]] => [1,2,2,1] => [2,2,1,1] => 4

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

[1,0,1,1,1,0,1,0,0,0] => [[2,2,2,1],[]] => [2,2,2,1] => [2,2,2,1] => 8

[1,0,1,1,1,1,0,0,0,0] => [[3,3,1],[]] => [3,3,1] => [3,3,1] => 9

[1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => [1,1,1,2] => [2,1,1,1] => 2

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

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

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

[1,1,0,0,1,1,1,0,0,0] => [[3,3,2],[1,1]] => [2,2,2] => [2,2,2] => 8

[1,1,0,1,0,0,1,0,1,0] => [[3,3,3],[2,2]] => [1,1,3] => [3,1,1] => 3

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

[1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => [1,4] => [4,1] => 4

[1,1,0,1,0,1,0,1,0,0] => [[5],[]] => [5] => [5] => 5

[1,1,0,1,0,1,1,0,0,0] => [[4,4],[2]] => [2,4] => [4,2] => 8

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

[1,1,0,1,1,0,0,1,0,0] => [[4,3],[1]] => [3,3] => [3,3] => 9

[1,1,0,1,1,0,1,0,0,0] => [[3,3,3],[1,1]] => [2,2,3] => [3,2,2] => 12

[1,1,0,1,1,1,0,0,0,0] => [[4,4],[1]] => [3,4] => [4,3] => 12

[1,1,1,0,0,0,1,0,1,0] => [[2,2,2,2],[1,1]] => [1,1,2,2] => [2,2,1,1] => 4

[1,1,1,0,0,0,1,1,0,0] => [[3,2,2],[1]] => [2,2,2] => [2,2,2] => 8

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

[1,1,1,0,0,1,0,1,0,0] => [[4,2],[]] => [4,2] => [4,2] => 8

[1,1,1,0,0,1,1,0,0,0] => [[3,3,2],[1]] => [2,3,2] => [3,2,2] => 12

[1,1,1,0,1,0,0,0,1,0] => [[2,2,2,2],[1]] => [1,2,2,2] => [2,2,2,1] => 8

[1,1,1,0,1,0,0,1,0,0] => [[3,2,2],[]] => [3,2,2] => [3,2,2] => 12

[1,1,1,0,1,0,1,0,0,0] => [[2,2,2,2],[]] => [2,2,2,2] => [2,2,2,2] => 16

[1,1,1,0,1,1,0,0,0,0] => [[3,3,2],[]] => [3,3,2] => [3,3,2] => 18

[1,1,1,1,0,0,0,0,1,0] => [[3,3,3],[2]] => [1,3,3] => [3,3,1] => 9

[1,1,1,1,0,0,0,1,0,0] => [[4,3],[]] => [4,3] => [4,3] => 12

[1,1,1,1,0,0,1,0,0,0] => [[3,3,3],[1]] => [2,3,3] => [3,3,2] => 18

[1,1,1,1,0,1,0,0,0,0] => [[4,4],[]] => [4,4] => [4,4] => 16

[1,1,1,1,1,0,0,0,0,0] => [[3,3,3],[]] => [3,3,3] => [3,3,3] => 27

[1,0,1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1,1],[]] => [1,1,1,1,1,1] => [1,1,1,1,1,1] => 1

[1,0,1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1,1],[]] => [2,1,1,1,1] => [2,1,1,1,1] => 2

[1,0,1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1,1],[1]] => [1,2,1,1,1] => [2,1,1,1,1] => 2

[1,0,1,0,1,0,1,1,0,1,0,0] => [[3,1,1,1],[]] => [3,1,1,1] => [3,1,1,1] => 3

[1,0,1,0,1,0,1,1,1,0,0,0] => [[2,2,1,1,1],[]] => [2,2,1,1,1] => [2,2,1,1,1] => 4

[1,0,1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1,1],[1,1]] => [1,1,2,1,1] => [2,1,1,1,1] => 2

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

[1,0,1,0,1,1,0,1,0,0,1,0] => [[3,3,1,1],[2]] => [1,3,1,1] => [3,1,1,1] => 3

[1,0,1,0,1,1,0,1,0,1,0,0] => [[4,1,1],[]] => [4,1,1] => [4,1,1] => 4

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

[1,0,1,0,1,1,1,0,0,0,1,0] => [[2,2,2,1,1],[1]] => [1,2,2,1,1] => [2,2,1,1,1] => 4

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

[1,0,1,0,1,1,1,0,1,0,0,0] => [[2,2,2,1,1],[]] => [2,2,2,1,1] => [2,2,2,1,1] => 8

[1,0,1,0,1,1,1,1,0,0,0,0] => [[3,3,1,1],[]] => [3,3,1,1] => [3,3,1,1] => 9

[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] => [2,1,1,1,1] => 2

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

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

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

[1,0,1,1,0,0,1,1,1,0,0,0] => [[3,3,2,1],[1,1]] => [2,2,2,1] => [2,2,2,1] => 8

[1,0,1,1,0,1,0,0,1,0,1,0] => [[3,3,3,1],[2,2]] => [1,1,3,1] => [3,1,1,1] => 3

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

[1,0,1,1,0,1,0,1,0,0,1,0] => [[4,4,1],[3]] => [1,4,1] => [4,1,1] => 4

[1,0,1,1,0,1,0,1,0,1,0,0] => [[5,1],[]] => [5,1] => [5,1] => 5

[1,0,1,1,0,1,0,1,1,0,0,0] => [[4,4,1],[2]] => [2,4,1] => [4,2,1] => 8

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

[1,0,1,1,0,1,1,0,0,1,0,0] => [[4,3,1],[1]] => [3,3,1] => [3,3,1] => 9

[1,0,1,1,0,1,1,0,1,0,0,0] => [[3,3,3,1],[1,1]] => [2,2,3,1] => [3,2,2,1] => 12

[1,0,1,1,0,1,1,1,0,0,0,0] => [[4,4,1],[1]] => [3,4,1] => [4,3,1] => 12

[1,0,1,1,1,0,0,0,1,0,1,0] => [[2,2,2,2,1],[1,1]] => [1,1,2,2,1] => [2,2,1,1,1] => 4

[1,0,1,1,1,0,0,0,1,1,0,0] => [[3,2,2,1],[1]] => [2,2,2,1] => [2,2,2,1] => 8

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

[1,0,1,1,1,0,0,1,0,1,0,0] => [[4,2,1],[]] => [4,2,1] => [4,2,1] => 8

[1,0,1,1,1,0,0,1,1,0,0,0] => [[3,3,2,1],[1]] => [2,3,2,1] => [3,2,2,1] => 12

[1,0,1,1,1,0,1,0,0,0,1,0] => [[2,2,2,2,1],[1]] => [1,2,2,2,1] => [2,2,2,1,1] => 8

[1,0,1,1,1,0,1,0,0,1,0,0] => [[3,2,2,1],[]] => [3,2,2,1] => [3,2,2,1] => 12

[1,0,1,1,1,0,1,0,1,0,0,0] => [[2,2,2,2,1],[]] => [2,2,2,2,1] => [2,2,2,2,1] => 16

[1,0,1,1,1,0,1,1,0,0,0,0] => [[3,3,2,1],[]] => [3,3,2,1] => [3,3,2,1] => 18

[1,0,1,1,1,1,0,0,0,0,1,0] => [[3,3,3,1],[2]] => [1,3,3,1] => [3,3,1,1] => 9

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searching the database for the individual values of this statistic

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

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

$F_{4} = q + 3\ q^{2} + 2\ q^{3} + 4\ q^{4} + 2\ q^{6} + q^{8} + q^{9}$

$F_{5} = q + 4\ q^{2} + 3\ q^{3} + 8\ q^{4} + q^{5} + 6\ q^{6} + 6\ q^{8} + 3\ q^{9} + 5\ q^{12} + 2\ q^{16} + 2\ q^{18} + q^{27}$

Description

The product of the parts of an integer partition.

Map

to partition

Description

Sends a composition to the partition obtained by sorting the entries.

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

row lengths

Description

The sequence of row lengths from top to bottom.