Identifier
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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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
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
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.