- FindStat

Identifier

Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤ

Values

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

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$F_{1} = q$

$F_{2} = 2\ q$

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

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

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

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

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

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

Description

The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau.
A cylindrical tableau associated with a standard Young tableau

$T$ is the skew row-strict tableau obtained by gluing two copies of

$T$ such that the inner shape is a rectangle.
This statistic equals

$\max_C\big(\ell(C) - \ell(T)\big)$ , where

$\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.

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.