- FindStat

Identifier

Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[8] => [4,4] => [2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => 7

[7,1] => [4,3,1] => [3,2,2,1] => [[1,3,8],[2,5],[4,7],[6]] => 6

[6,2] => [3,3,1,1] => [4,2,2] => [[1,2,7,8],[3,4],[5,6]] => 5

[6,1,1] => [4,2,2] => [3,3,1,1] => [[1,4,5],[2,7,8],[3],[6]] => 6

[5,3] => [2,2,2,1,1] => [5,3] => [[1,2,3,7,8],[4,5,6]] => 4

[5,2,1] => [4,1,1,1,1] => [5,1,1,1] => [[1,5,6,7,8],[2],[3],[4]] => 4

[5,1,1,1] => [2,2,2,2] => [4,4] => [[1,2,3,4],[5,6,7,8]] => 5

[4,4] => [4,2,1,1] => [4,2,1,1] => [[1,4,7,8],[2,6],[3],[5]] => 5

[4,3,1] => [1,1,1,1,1,1,1,1] => [8] => [[1,2,3,4,5,6,7,8]] => 1

[4,2,2] => [2,1,1,1,1,1,1] => [7,1] => [[1,3,4,5,6,7,8],[2]] => 2

[4,2,1,1] => [3,1,1,1,1,1] => [6,1,1] => [[1,4,5,6,7,8],[2],[3]] => 3

[4,1,1,1,1] => [5,2,1] => [3,2,1,1,1] => [[1,5,8],[2,7],[3],[4],[6]] => 6

[3,3,2] => [2,2,1,1,1,1] => [6,2] => [[1,2,5,6,7,8],[3,4]] => 3

[3,3,1,1] => [3,2,1,1,1] => [5,2,1] => [[1,3,6,7,8],[2,5],[4]] => 4

[3,2,2,1] => [3,2,2,1] => [4,3,1] => [[1,3,4,8],[2,6,7],[5]] => 5

[3,2,1,1,1] => [5,1,1,1] => [4,1,1,1,1] => [[1,6,7,8],[2],[3],[4],[5]] => 5

[3,1,1,1,1,1] => [6,1,1] => [3,1,1,1,1,1] => [[1,7,8],[2],[3],[4],[5],[6]] => 6

[2,2,2,2] => [3,3,2] => [3,3,2] => [[1,2,5],[3,4,8],[6,7]] => 6

[2,2,2,1,1] => [5,3] => [2,2,2,1,1] => [[1,4],[2,6],[3,8],[5],[7]] => 7

[2,2,1,1,1,1] => [6,2] => [2,2,1,1,1,1] => [[1,6],[2,8],[3],[4],[5],[7]] => 7

[2,1,1,1,1,1,1] => [7,1] => [2,1,1,1,1,1,1] => [[1,8],[2],[3],[4],[5],[6],[7]] => 7

[1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => 8

[9] => [5,4] => [2,2,2,2,1] => [[1,3],[2,5],[4,7],[6,9],[8]] => 8

[8,1] => [4,4,1] => [3,2,2,2] => [[1,2,9],[3,4],[5,6],[7,8]] => 7

[7,2] => [4,3,1,1] => [4,2,2,1] => [[1,3,8,9],[2,5],[4,7],[6]] => 6

[7,1,1] => [4,3,2] => [3,3,2,1] => [[1,3,6],[2,5,9],[4,8],[7]] => 7

[6,3] => [4,2,2,1] => [4,3,1,1] => [[1,4,5,9],[2,7,8],[3],[6]] => 6

[6,2,1] => [4,2,1,1,1] => [5,2,1,1] => [[1,4,7,8,9],[2,6],[3],[5]] => 5

[6,1,1,1] => [5,2,2] => [3,3,1,1,1] => [[1,5,6],[2,8,9],[3],[4],[7]] => 7

[5,4] => [2,2,2,2,1] => [5,4] => [[1,2,3,4,9],[5,6,7,8]] => 5

[5,3,1] => [3,1,1,1,1,1,1] => [7,1,1] => [[1,4,5,6,7,8,9],[2],[3]] => 3

[5,2,2] => [3,2,1,1,1,1] => [6,2,1] => [[1,3,6,7,8,9],[2,5],[4]] => 4

[5,2,1,1] => [3,3,1,1,1] => [5,2,2] => [[1,2,7,8,9],[3,4],[5,6]] => 5

[5,1,1,1,1] => [3,2,2,2] => [4,4,1] => [[1,3,4,5],[2,7,8,9],[6]] => 6

[4,4,1] => [4,1,1,1,1,1] => [6,1,1,1] => [[1,5,6,7,8,9],[2],[3],[4]] => 4

[4,3,2] => [1,1,1,1,1,1,1,1,1] => [9] => [[1,2,3,4,5,6,7,8,9]] => 1

[4,3,1,1] => [2,1,1,1,1,1,1,1] => [8,1] => [[1,3,4,5,6,7,8,9],[2]] => 2

[4,2,2,1] => [2,2,1,1,1,1,1] => [7,2] => [[1,2,5,6,7,8,9],[3,4]] => 3

[4,2,1,1,1] => [5,1,1,1,1] => [5,1,1,1,1] => [[1,6,7,8,9],[2],[3],[4],[5]] => 5

[4,1,1,1,1,1] => [6,2,1] => [3,2,1,1,1,1] => [[1,6,9],[2,8],[3],[4],[5],[7]] => 7

[3,3,3] => [3,2,2,1,1] => [5,3,1] => [[1,3,4,8,9],[2,6,7],[5]] => 5

[3,3,2,1] => [2,2,2,1,1,1] => [6,3] => [[1,2,3,7,8,9],[4,5,6]] => 4

[3,3,1,1,1] => [5,2,1,1] => [4,2,1,1,1] => [[1,5,8,9],[2,7],[3],[4],[6]] => 6

[3,2,2,2] => [3,3,2,1] => [4,3,2] => [[1,2,5,9],[3,4,8],[6,7]] => 6

[3,2,2,1,1] => [5,3,1] => [3,2,2,1,1] => [[1,4,9],[2,6],[3,8],[5],[7]] => 7

[3,2,1,1,1,1] => [6,1,1,1] => [4,1,1,1,1,1] => [[1,7,8,9],[2],[3],[4],[5],[6]] => 6

[3,1,1,1,1,1,1] => [7,1,1] => [3,1,1,1,1,1,1] => [[1,8,9],[2],[3],[4],[5],[6],[7]] => 7

[2,2,2,2,1] => [3,3,3] => [3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => 7

[2,2,2,1,1,1] => [6,3] => [2,2,2,1,1,1] => [[1,5],[2,7],[3,9],[4],[6],[8]] => 8

[2,2,1,1,1,1,1] => [7,2] => [2,2,1,1,1,1,1] => [[1,7],[2,9],[3],[4],[5],[6],[8]] => 8

[2,1,1,1,1,1,1,1] => [8,1] => [2,1,1,1,1,1,1,1] => [[1,9],[2],[3],[4],[5],[6],[7],[8]] => 8

[1,1,1,1,1,1,1,1,1] => [9] => [1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => 9

[10] => [5,5] => [2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => 9

[9,1] => [5,4,1] => [3,2,2,2,1] => [[1,3,10],[2,5],[4,7],[6,9],[8]] => 8

[8,2] => [4,4,1,1] => [4,2,2,2] => [[1,2,9,10],[3,4],[5,6],[7,8]] => 7

[8,1,1] => [4,4,2] => [3,3,2,2] => [[1,2,7],[3,4,10],[5,6],[8,9]] => 8

[7,3] => [4,3,2,1] => [4,3,2,1] => [[1,3,6,10],[2,5,9],[4,8],[7]] => 7

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$F_{1} = q$

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

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

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

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

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

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

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

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

$F_{10} = q + q^{2} + 2\ q^{3} + 3\ q^{4} + 5\ q^{5} + 7\ q^{6} + 9\ q^{7} + 8\ q^{8} + 5\ q^{9} + q^{10}$

Description

The first entry in the last row of a standard tableau.
For the last entry in the first row, see St000734The last entry in the first row of a standard tableau..

Map

Loehr-Warrington inverse

Description

Return a partition whose length is the diagonal inversion number of the preimage.

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.