Identifier
Values
[[1]] => [[1]] => [1] => [[1],[]] => 1
[[1,2]] => [[1,2]] => [2] => [[2],[]] => 2
[[1],[2]] => [[1],[2]] => [2] => [[2],[]] => 2
[[1,2,3]] => [[1,2,3]] => [3] => [[3],[]] => 2
[[1,3],[2]] => [[1,2],[3]] => [2,1] => [[2,2],[1]] => 3
[[1,2],[3]] => [[1,3],[2]] => [3] => [[3],[]] => 2
[[1],[2],[3]] => [[1],[2],[3]] => [3] => [[3],[]] => 2
[[1,2,3,4]] => [[1,2,3,4]] => [4] => [[4],[]] => 2
[[1,3,4],[2]] => [[1,2,3],[4]] => [3,1] => [[3,3],[2]] => 3
[[1,2,4],[3]] => [[1,2,4],[3]] => [2,2] => [[3,2],[1]] => 4
[[1,2,3],[4]] => [[1,3,4],[2]] => [4] => [[4],[]] => 2
[[1,3],[2,4]] => [[1,3],[2,4]] => [3,1] => [[3,3],[2]] => 3
[[1,2],[3,4]] => [[1,2],[3,4]] => [2,2] => [[3,2],[1]] => 4
[[1,4],[2],[3]] => [[1,2],[3],[4]] => [2,2] => [[3,2],[1]] => 4
[[1,3],[2],[4]] => [[1,3],[2],[4]] => [3,1] => [[3,3],[2]] => 3
[[1,2],[3],[4]] => [[1,4],[2],[3]] => [4] => [[4],[]] => 2
[[1],[2],[3],[4]] => [[1],[2],[3],[4]] => [4] => [[4],[]] => 2
[[1,2,3,4,5]] => [[1,2,3,4,5]] => [5] => [[5],[]] => 2
[[1,3,4,5],[2]] => [[1,2,3,4],[5]] => [4,1] => [[4,4],[3]] => 3
[[1,2,4,5],[3]] => [[1,2,3,5],[4]] => [3,2] => [[4,3],[2]] => 4
[[1,2,3,5],[4]] => [[1,2,4,5],[3]] => [2,3] => [[4,2],[1]] => 4
[[1,2,3,4],[5]] => [[1,3,4,5],[2]] => [5] => [[5],[]] => 2
[[1,3,5],[2,4]] => [[1,2,4],[3,5]] => [2,2,1] => [[3,3,2],[2,1]] => 5
[[1,2,5],[3,4]] => [[1,2,3],[4,5]] => [3,2] => [[4,3],[2]] => 4
[[1,3,4],[2,5]] => [[1,3,4],[2,5]] => [4,1] => [[4,4],[3]] => 3
[[1,2,4],[3,5]] => [[1,3,5],[2,4]] => [3,2] => [[4,3],[2]] => 4
[[1,2,3],[4,5]] => [[1,2,5],[3,4]] => [2,3] => [[4,2],[1]] => 4
[[1,4,5],[2],[3]] => [[1,2,3],[4],[5]] => [3,2] => [[4,3],[2]] => 4
[[1,3,5],[2],[4]] => [[1,2,4],[3],[5]] => [2,2,1] => [[3,3,2],[2,1]] => 5
[[1,2,5],[3],[4]] => [[1,2,5],[3],[4]] => [2,3] => [[4,2],[1]] => 4
[[1,3,4],[2],[5]] => [[1,3,4],[2],[5]] => [4,1] => [[4,4],[3]] => 3
[[1,2,4],[3],[5]] => [[1,3,5],[2],[4]] => [3,2] => [[4,3],[2]] => 4
[[1,2,3],[4],[5]] => [[1,4,5],[2],[3]] => [5] => [[5],[]] => 2
[[1,4],[2,5],[3]] => [[1,3],[2,4],[5]] => [3,2] => [[4,3],[2]] => 4
[[1,3],[2,5],[4]] => [[1,2],[3,4],[5]] => [2,2,1] => [[3,3,2],[2,1]] => 5
[[1,2],[3,5],[4]] => [[1,2],[3,5],[4]] => [2,3] => [[4,2],[1]] => 4
[[1,3],[2,4],[5]] => [[1,4],[2,5],[3]] => [4,1] => [[4,4],[3]] => 3
[[1,2],[3,4],[5]] => [[1,3],[2,5],[4]] => [3,2] => [[4,3],[2]] => 4
[[1,5],[2],[3],[4]] => [[1,2],[3],[4],[5]] => [2,3] => [[4,2],[1]] => 4
[[1,4],[2],[3],[5]] => [[1,3],[2],[4],[5]] => [3,2] => [[4,3],[2]] => 4
[[1,3],[2],[4],[5]] => [[1,4],[2],[3],[5]] => [4,1] => [[4,4],[3]] => 3
[[1,2],[3],[4],[5]] => [[1,5],[2],[3],[4]] => [5] => [[5],[]] => 2
[[1],[2],[3],[4],[5]] => [[1],[2],[3],[4],[5]] => [5] => [[5],[]] => 2
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Description
The number of corners of a skew partition.
This is also known as the number of removable cells of the skew partition.
Map
to ribbon
Description
The ribbon shape corresponding to an integer composition.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
Map
peak composition
Description
The composition corresponding to the peak set of a standard tableau.
Let $T$ be a standard tableau of size $n$.
An entry $i$ of $T$ is a descent if $i+1$ is in a lower row (in English notation), otherwise $i$ is an ascent.
An entry $2 \leq i \leq n-1$ is a peak, if $i-1$ is an ascent and $i$ is a descent.
This map returns the composition $c_1,\dots,c_k$ of $n$ such that $\{c_1, c_1+c_2,\dots, c_1+\dots+c_k\}$ is the peak set of $T$.
Map
Schützenberger involution
Description
Sends a standard tableau to the standard tableau obtained via the Schützenberger involution.