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Identifier

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤ

Values

{{1}} => [1] => [1] => [[1]] => 0

{{1,2}} => [2] => [1,1] => [[1],[2]] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1,2,3,4,5}} => [5] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => 10

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1},{2},{3},{4},{5}} => [1,1,1,1,1] => [5] => [[1,2,3,4,5]] => 0

{{1,2,3,4,5,6}} => [6] => [1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => 15

{{1,2,3,4,5},{6}} => [5,1] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => 10

{{1,2,3,4,6},{5}} => [5,1] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => 10

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

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

{{1,2,3,5,6},{4}} => [5,1] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => 10

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

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

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

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

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

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

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

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

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

{{1,2,4,5,6},{3}} => [5,1] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => 10

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 3\ q + q^{3}$

$F_{4} = 1 + 6\ q + 3\ q^{2} + 4\ q^{3} + q^{6}$

$F_{5} = 1 + 10\ q + 15\ q^{2} + 10\ q^{3} + 10\ q^{4} + 5\ q^{6} + q^{10}$

$F_{6} = 1 + 15\ q + 45\ q^{2} + 35\ q^{3} + 60\ q^{4} + 25\ q^{6} + 15\ q^{7} + 6\ q^{10} + q^{15}$

$F_{7} = 1 + 21\ q + 105\ q^{2} + 140\ q^{3} + 210\ q^{4} + 105\ q^{5} + 105\ q^{6} + 105\ q^{7} + 35\ q^{9} + 21\ q^{10} + 21\ q^{11} + 7\ q^{15} + q^{21}$

Description

The (standard) major index of a standard tableau.
A descent of a standard tableau

$T$ is an index

$i$ such that

$i+1$ appears in a row strictly below the row of

$i$ . The (standard) major index is the the sum of the descents.

Map

conjugate

Description

Return the conjugate partition of the partition.
The conjugate partition of the partition

$\lambda$ of

$n$ is the partition

$\lambda^*$ whose Ferrers diagram is obtained from the diagram of

$\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.

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.

Map

shape

Description

Sends a set partition to the integer partition obtained by the sizes of the blocks.