- FindStat

Identifier

Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000899: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [1,1,1,1,1,1,1] => 7

[1,1,1,1,1,2] => [2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => [1,1,1,1,1,2] => 5

[1,1,1,1,2,1] => [2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => [1,1,1,1,1,2] => 5

[1,1,1,1,3] => [3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => [1,1,1,1,3] => 4

[1,1,1,2,1,1] => [2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => [1,1,1,1,1,2] => 5

[1,1,1,2,2] => [2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => [1,1,1,2,2] => 3

[1,1,1,3,1] => [3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => [1,1,1,1,3] => 4

[1,1,1,4] => [4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => [1,1,1,4] => 3

[1,1,2,1,1,1] => [2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => [1,1,1,1,1,2] => 5

[1,1,2,1,2] => [2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => [1,1,1,2,2] => 3

[1,1,2,2,1] => [2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => [1,1,1,2,2] => 3

[1,1,2,3] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => [1,1,2,3] => 2

[1,1,3,1,1] => [3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => [1,1,1,1,3] => 4

[1,1,3,2] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => [1,1,2,3] => 2

[1,1,4,1] => [4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => [1,1,1,4] => 3

[1,1,5] => [5,1,1] => [[1,4,5,6,7],[2],[3]] => [1,1,5] => 2

[1,2,1,1,1,1] => [2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => [1,1,1,1,1,2] => 5

[1,2,1,1,2] => [2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => [1,1,1,2,2] => 3

[1,2,1,2,1] => [2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => [1,1,1,2,2] => 3

[1,2,1,3] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => [1,1,2,3] => 2

[1,2,2,1,1] => [2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => [1,1,1,2,2] => 3

[1,2,2,2] => [2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => [1,2,2,2] => 3

[1,2,3,1] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => [1,1,2,3] => 2

[1,2,4] => [4,2,1] => [[1,3,6,7],[2,5],[4]] => [1,2,4] => 1

[1,3,1,1,1] => [3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => [1,1,1,1,3] => 4

[1,3,1,2] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => [1,1,2,3] => 2

[1,3,2,1] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => [1,1,2,3] => 2

[1,3,3] => [3,3,1] => [[1,3,4],[2,6,7],[5]] => [1,3,3] => 2

[1,4,1,1] => [4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => [1,1,1,4] => 3

[1,4,2] => [4,2,1] => [[1,3,6,7],[2,5],[4]] => [1,2,4] => 1

[1,5,1] => [5,1,1] => [[1,4,5,6,7],[2],[3]] => [1,1,5] => 2

[1,6] => [6,1] => [[1,3,4,5,6,7],[2]] => [1,6] => 1

[2,1,1,1,1,1] => [2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => [1,1,1,1,1,2] => 5

[2,1,1,1,2] => [2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => [1,1,1,2,2] => 3

[2,1,1,2,1] => [2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => [1,1,1,2,2] => 3

[2,1,1,3] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => [1,1,2,3] => 2

[2,1,2,1,1] => [2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => [1,1,1,2,2] => 3

[2,1,2,2] => [2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => [1,2,2,2] => 3

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$F_{1} = q$

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

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

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

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

$F_{6} = 11\ q + 10\ q^{2} + 5\ q^{3} + 5\ q^{4} + q^{6}$

$F_{7} = 13\ q + 21\ q^{2} + 18\ q^{3} + 5\ q^{4} + 6\ q^{5} + q^{7}$

$F_{8} = 19\ q + 40\ q^{2} + 34\ q^{3} + 21\ q^{4} + 6\ q^{5} + 7\ q^{6} + q^{8}$

$F_{9} = 27\ q + 87\ q^{2} + 59\ q^{3} + 40\ q^{4} + 27\ q^{5} + 7\ q^{6} + 8\ q^{7} + q^{9}$

Description

The maximal number of repetitions of an integer composition.
This is the maximal part of the composition obtained by applying the delta morphism.

Map

to partition

Description

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

Map

horizontal strip sizes

Description

The composition of horizontal strip sizes.
We associate to a standard Young tableau

$T$ the composition

$(c_1,\dots,c_k)$ , such that

$k$ is minimal and the numbers

$c_1+\dots+c_i + 1,\dots,c_1+\dots+c_{i+1}$ form a horizontal strip in

$T$ for all

$i$ .

Map

reading tableau

Description

Return the RSK recording tableau of the reading word of the (standard) tableau

$T$ labeled down (in English convention) each column to the shape of a partition.