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Identifier

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001697: 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] => [[1,2],[3]] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1,2,5,6},{3,4}} => [4,2] => [2,2,1,1] => [[1,2],[3,4],[5],[6]] => 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 shifted natural comajor index of a standard Young tableau.
A natural descent of a standard tableau

$T$ is an entry

$i$ such that

$i+1$ appears in a higher row than

$i$ in English notation.
The natural comajor index of a tableau of shape

$\lambda$ , size

$n$ with natural descent set

$D$ is then

$b(\lambda)+\sum_{d\in D} n-d$ , where

$b(\lambda) = \sum_i (i-1)\lambda_i$ .

Map

shape

Description

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

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

initial tableau

Description

Sends an integer partition to the standard tableau obtained by filling the numbers

$1$ through

$n$ row by row.