- FindStat

Identifier

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St001933: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,1,1,0,0,0,0,0] => [5] => [[5],[]] => [5] => 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],[]] => [1,1,1,1,1,1] => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

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

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

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

$F_{5} = 24\ q + 12\ q^{2} + 4\ q^{3} + q^{4} + q^{5}$

$F_{6} = 75\ q + 38\ q^{2} + 13\ q^{3} + 4\ q^{4} + q^{5} + q^{6}$

$F_{7} = 243\ q + 125\ q^{2} + 42\ q^{3} + 13\ q^{4} + 4\ q^{5} + q^{6} + q^{7}$

Description

The largest multiplicity of a part in an integer partition.

Map

to ribbon

Description

The ribbon shape corresponding to an integer composition.
For an integer composition

$(a_1, \dots, a_n)$ , this is the ribbon shape whose

$i$ th row from the bottom has

$a_i$ cells.

Map

outer shape

Description

The outer shape of the skew partition.

Map

touch composition

Description

Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.