- FindStat

Identifier

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000108: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,2,1] => [[3,3,2],[2,1]] => [2,1] => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,2,1] => [[3,3,2,1],[2,1]] => [2,1] => 5

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

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

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

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

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

[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 5

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

[2,1,2,1] => [[3,3,2,2],[2,1,1]] => [2,1,1] => 7

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

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

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

[2,3,1] => [[4,4,2],[3,1]] => [3,1] => 7

[2,4] => [[5,2],[1]] => [1] => 2

[3,1,1,1] => [[3,3,3,3],[2,2,2]] => [2,2,2] => 10

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

[3,2,1] => [[4,4,3],[3,2]] => [3,2] => 9

[3,3] => [[5,3],[2]] => [2] => 3

[4,1,1] => [[4,4,4],[3,3]] => [3,3] => 10

[4,2] => [[5,4],[3]] => [3] => 4

[5,1] => [[5,5],[4]] => [4] => 5

[6] => [[6],[]] => [] => 1

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

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

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

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

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

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

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

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

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

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

[1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => [2,1] => 5

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

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

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

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

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

[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => 5

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

[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => [2,1,1] => 7

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

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

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

[1,2,3,1] => [[4,4,2,1],[3,1]] => [3,1] => 7

[1,2,4] => [[5,2,1],[1]] => [1] => 2

[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => [2,2,2] => 10

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

[1,3,2,1] => [[4,4,3,1],[3,2]] => [3,2] => 9

[1,3,3] => [[5,3,1],[2]] => [2] => 3

[1,4,1,1] => [[4,4,4,1],[3,3]] => [3,3] => 10

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

[1,5,1] => [[5,5,1],[4]] => [4] => 5

[1,6] => [[6,1],[]] => [] => 1

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

[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 5

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

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

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

[2,1,2,2] => [[4,3,2,2],[2,1,1]] => [2,1,1] => 7

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$F_{1} = q$

$F_{2} = 2\ q$

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

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

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

$F_{6} = 6\ q + 4\ q^{2} + 6\ q^{3} + 4\ q^{4} + 4\ q^{5} + 2\ q^{6} + 2\ q^{7} + 2\ q^{9} + 2\ q^{10}$

$F_{7} = 7\ q + 5\ q^{2} + 8\ q^{3} + 6\ q^{4} + 7\ q^{5} + 5\ q^{6} + 4\ q^{7} + 6\ q^{9} + 5\ q^{10} + 2\ q^{12} + 3\ q^{14} + 2\ q^{15} + 2\ q^{16} + q^{19} + q^{20}$

Description

The number of partitions contained in the given 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

inner shape

Description

The inner shape of a skew partition.