- FindStat

Identifier

Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤ

Values

[1] => [1] => [1] => [[1]] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,1,1,1] => [1,1,1,1,1,1] => [6] => [[1,2,3,4,5,6]] => 0

[1,1,1,1,2] => [2,1,1,1,1] => [5,1] => [[1,3,4,5,6],[2]] => 0

[1,1,1,2,1] => [2,1,1,1,1] => [5,1] => [[1,3,4,5,6],[2]] => 0

[1,1,1,3] => [3,1,1,1] => [4,1,1] => [[1,4,5,6],[2],[3]] => 0

[1,1,2,1,1] => [2,1,1,1,1] => [5,1] => [[1,3,4,5,6],[2]] => 0

[1,1,2,2] => [2,2,1,1] => [4,2] => [[1,2,5,6],[3,4]] => 0

[1,1,3,1] => [3,1,1,1] => [4,1,1] => [[1,4,5,6],[2],[3]] => 0

[1,1,4] => [4,1,1] => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => 0

[1,2,1,1,1] => [2,1,1,1,1] => [5,1] => [[1,3,4,5,6],[2]] => 0

[1,2,1,2] => [2,2,1,1] => [4,2] => [[1,2,5,6],[3,4]] => 0

[1,2,2,1] => [2,2,1,1] => [4,2] => [[1,2,5,6],[3,4]] => 0

[1,2,3] => [3,2,1] => [3,2,1] => [[1,3,6],[2,5],[4]] => 0

[1,3,1,1] => [3,1,1,1] => [4,1,1] => [[1,4,5,6],[2],[3]] => 0

[1,3,2] => [3,2,1] => [3,2,1] => [[1,3,6],[2,5],[4]] => 0

[1,4,1] => [4,1,1] => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => 0

[1,5] => [5,1] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => 0

[2,1,1,1,1] => [2,1,1,1,1] => [5,1] => [[1,3,4,5,6],[2]] => 0

[2,1,1,2] => [2,2,1,1] => [4,2] => [[1,2,5,6],[3,4]] => 0

[2,1,2,1] => [2,2,1,1] => [4,2] => [[1,2,5,6],[3,4]] => 0

[2,1,3] => [3,2,1] => [3,2,1] => [[1,3,6],[2,5],[4]] => 0

[2,2,1,1] => [2,2,1,1] => [4,2] => [[1,2,5,6],[3,4]] => 0

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

[2,3,1] => [3,2,1] => [3,2,1] => [[1,3,6],[2,5],[4]] => 0

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

[3,1,1,1] => [3,1,1,1] => [4,1,1] => [[1,4,5,6],[2],[3]] => 0

[3,1,2] => [3,2,1] => [3,2,1] => [[1,3,6],[2,5],[4]] => 0

[3,2,1] => [3,2,1] => [3,2,1] => [[1,3,6],[2,5],[4]] => 0

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

[4,1,1] => [4,1,1] => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => 0

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

[5,1] => [5,1] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => 0

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

[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [7] => [[1,2,3,4,5,6,7]] => 0

[1,1,1,1,1,2] => [2,1,1,1,1,1] => [6,1] => [[1,3,4,5,6,7],[2]] => 0

[1,1,1,1,2,1] => [2,1,1,1,1,1] => [6,1] => [[1,3,4,5,6,7],[2]] => 0

[1,1,1,1,3] => [3,1,1,1,1] => [5,1,1] => [[1,4,5,6,7],[2],[3]] => 0

[1,1,1,2,1,1] => [2,1,1,1,1,1] => [6,1] => [[1,3,4,5,6,7],[2]] => 0

[1,1,1,2,2] => [2,2,1,1,1] => [5,2] => [[1,2,5,6,7],[3,4]] => 0

[1,1,1,3,1] => [3,1,1,1,1] => [5,1,1] => [[1,4,5,6,7],[2],[3]] => 0

[1,1,1,4] => [4,1,1,1] => [4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => 0

[1,1,2,1,1,1] => [2,1,1,1,1,1] => [6,1] => [[1,3,4,5,6,7],[2]] => 0

[1,1,2,1,2] => [2,2,1,1,1] => [5,2] => [[1,2,5,6,7],[3,4]] => 0

[1,1,2,2,1] => [2,2,1,1,1] => [5,2] => [[1,2,5,6,7],[3,4]] => 0

[1,1,2,3] => [3,2,1,1] => [4,2,1] => [[1,3,6,7],[2,5],[4]] => 0

[1,1,3,1,1] => [3,1,1,1,1] => [5,1,1] => [[1,4,5,6,7],[2],[3]] => 0

[1,1,3,2] => [3,2,1,1] => [4,2,1] => [[1,3,6,7],[2,5],[4]] => 0

[1,1,4,1] => [4,1,1,1] => [4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => 0

[1,1,5] => [5,1,1] => [3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => 0

[1,2,1,1,1,1] => [2,1,1,1,1,1] => [6,1] => [[1,3,4,5,6,7],[2]] => 0

[1,2,1,1,2] => [2,2,1,1,1] => [5,2] => [[1,2,5,6,7],[3,4]] => 0

[1,2,1,2,1] => [2,2,1,1,1] => [5,2] => [[1,2,5,6,7],[3,4]] => 0

[1,2,1,3] => [3,2,1,1] => [4,2,1] => [[1,3,6,7],[2,5],[4]] => 0

[1,2,2,1,1] => [2,2,1,1,1] => [5,2] => [[1,2,5,6,7],[3,4]] => 0

[1,2,2,2] => [2,2,2,1] => [4,3] => [[1,2,3,7],[4,5,6]] => 0

[1,2,3,1] => [3,2,1,1] => [4,2,1] => [[1,3,6,7],[2,5],[4]] => 0

[1,2,4] => [4,2,1] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => 0

[1,3,1,1,1] => [3,1,1,1,1] => [5,1,1] => [[1,4,5,6,7],[2],[3]] => 0

[1,3,1,2] => [3,2,1,1] => [4,2,1] => [[1,3,6,7],[2,5],[4]] => 0

[1,3,2,1] => [3,2,1,1] => [4,2,1] => [[1,3,6,7],[2,5],[4]] => 0

[1,3,3] => [3,3,1] => [3,2,2] => [[1,2,7],[3,4],[5,6]] => 0

[1,4,1,1] => [4,1,1,1] => [4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => 0

[1,4,2] => [4,2,1] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => 0

[1,5,1] => [5,1,1] => [3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => 0

[1,6] => [6,1] => [2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => 0

[2,1,1,1,1,1] => [2,1,1,1,1,1] => [6,1] => [[1,3,4,5,6,7],[2]] => 0

[2,1,1,1,2] => [2,2,1,1,1] => [5,2] => [[1,2,5,6,7],[3,4]] => 0

[2,1,1,2,1] => [2,2,1,1,1] => [5,2] => [[1,2,5,6,7],[3,4]] => 0

[2,1,1,3] => [3,2,1,1] => [4,2,1] => [[1,3,6,7],[2,5],[4]] => 0

[2,1,2,1,1] => [2,2,1,1,1] => [5,2] => [[1,2,5,6,7],[3,4]] => 0

[2,1,2,2] => [2,2,2,1] => [4,3] => [[1,2,3,7],[4,5,6]] => 0

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$F_{1} = 1$

$F_{2} = 1 + q$

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

$F_{4} = 6 + q + q^{3}$

$F_{5} = 13 + 2\ q + q^{4}$

$F_{6} = 27 + 3\ q + q^{2} + q^{5}$

$F_{7} = 56 + 5\ q + 2\ q^{2} + q^{6}$

$F_{8} = 115 + 9\ q + 2\ q^{2} + q^{3} + q^{7}$

Description

The maximal overlap of the cylindrical tableau associated with 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.
The overlap, recorded in this statistic, equals

$\max_C\big(2\ell(T) - \ell(C)\big)$ , where

$\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
In particular, the statistic equals

$0$ , if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.

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

to partition

Description

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

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.