Identifier
-
Mp00295:
Standard tableaux
—valley composition⟶
Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000089: Integer compositions ⟶ ℤ
Values
[[1]] => [1] => [1] => 0
[[1,2]] => [2] => [1,1] => 0
[[1],[2]] => [2] => [1,1] => 0
[[1,2,3]] => [3] => [1,1,1] => 0
[[1,3],[2]] => [2,1] => [2,1] => 1
[[1,2],[3]] => [3] => [1,1,1] => 0
[[1],[2],[3]] => [3] => [1,1,1] => 0
[[1,2,3,4]] => [4] => [1,1,1,1] => 0
[[1,3,4],[2]] => [2,2] => [1,2,1] => 2
[[1,2,4],[3]] => [3,1] => [2,1,1] => 1
[[1,2,3],[4]] => [4] => [1,1,1,1] => 0
[[1,3],[2,4]] => [2,2] => [1,2,1] => 2
[[1,2],[3,4]] => [3,1] => [2,1,1] => 1
[[1,4],[2],[3]] => [3,1] => [2,1,1] => 1
[[1,3],[2],[4]] => [2,2] => [1,2,1] => 2
[[1,2],[3],[4]] => [4] => [1,1,1,1] => 0
[[1],[2],[3],[4]] => [4] => [1,1,1,1] => 0
[[1,2,3,4,5]] => [5] => [1,1,1,1,1] => 0
[[1,3,4,5],[2]] => [2,3] => [1,1,2,1] => 2
[[1,2,4,5],[3]] => [3,2] => [1,2,1,1] => 2
[[1,2,3,5],[4]] => [4,1] => [2,1,1,1] => 1
[[1,2,3,4],[5]] => [5] => [1,1,1,1,1] => 0
[[1,3,5],[2,4]] => [2,2,1] => [2,2,1] => 1
[[1,2,5],[3,4]] => [3,2] => [1,2,1,1] => 2
[[1,3,4],[2,5]] => [2,3] => [1,1,2,1] => 2
[[1,2,4],[3,5]] => [3,2] => [1,2,1,1] => 2
[[1,2,3],[4,5]] => [4,1] => [2,1,1,1] => 1
[[1,4,5],[2],[3]] => [3,2] => [1,2,1,1] => 2
[[1,3,5],[2],[4]] => [2,2,1] => [2,2,1] => 1
[[1,2,5],[3],[4]] => [4,1] => [2,1,1,1] => 1
[[1,3,4],[2],[5]] => [2,3] => [1,1,2,1] => 2
[[1,2,4],[3],[5]] => [3,2] => [1,2,1,1] => 2
[[1,2,3],[4],[5]] => [5] => [1,1,1,1,1] => 0
[[1,4],[2,5],[3]] => [3,2] => [1,2,1,1] => 2
[[1,3],[2,5],[4]] => [2,2,1] => [2,2,1] => 1
[[1,2],[3,5],[4]] => [4,1] => [2,1,1,1] => 1
[[1,3],[2,4],[5]] => [2,3] => [1,1,2,1] => 2
[[1,2],[3,4],[5]] => [3,2] => [1,2,1,1] => 2
[[1,5],[2],[3],[4]] => [4,1] => [2,1,1,1] => 1
[[1,4],[2],[3],[5]] => [3,2] => [1,2,1,1] => 2
[[1,3],[2],[4],[5]] => [2,3] => [1,1,2,1] => 2
[[1,2],[3],[4],[5]] => [5] => [1,1,1,1,1] => 0
[[1],[2],[3],[4],[5]] => [5] => [1,1,1,1,1] => 0
[[1,2,3,4,5,6]] => [6] => [1,1,1,1,1,1] => 0
[[1,3,4,5,6],[2]] => [2,4] => [1,1,1,2,1] => 2
[[1,2,4,5,6],[3]] => [3,3] => [1,1,2,1,1] => 2
[[1,2,3,5,6],[4]] => [4,2] => [1,2,1,1,1] => 2
[[1,2,3,4,6],[5]] => [5,1] => [2,1,1,1,1] => 1
[[1,2,3,4,5],[6]] => [6] => [1,1,1,1,1,1] => 0
[[1,3,5,6],[2,4]] => [2,2,2] => [1,2,2,1] => 2
[[1,2,5,6],[3,4]] => [3,3] => [1,1,2,1,1] => 2
[[1,3,4,6],[2,5]] => [2,3,1] => [2,1,2,1] => 3
[[1,2,4,6],[3,5]] => [3,2,1] => [2,2,1,1] => 1
[[1,2,3,6],[4,5]] => [4,2] => [1,2,1,1,1] => 2
[[1,3,4,5],[2,6]] => [2,4] => [1,1,1,2,1] => 2
[[1,2,4,5],[3,6]] => [3,3] => [1,1,2,1,1] => 2
[[1,2,3,5],[4,6]] => [4,2] => [1,2,1,1,1] => 2
[[1,2,3,4],[5,6]] => [5,1] => [2,1,1,1,1] => 1
[[1,4,5,6],[2],[3]] => [3,3] => [1,1,2,1,1] => 2
[[1,3,5,6],[2],[4]] => [2,2,2] => [1,2,2,1] => 2
[[1,2,5,6],[3],[4]] => [4,2] => [1,2,1,1,1] => 2
[[1,3,4,6],[2],[5]] => [2,3,1] => [2,1,2,1] => 3
[[1,2,4,6],[3],[5]] => [3,2,1] => [2,2,1,1] => 1
[[1,2,3,6],[4],[5]] => [5,1] => [2,1,1,1,1] => 1
[[1,3,4,5],[2],[6]] => [2,4] => [1,1,1,2,1] => 2
[[1,2,4,5],[3],[6]] => [3,3] => [1,1,2,1,1] => 2
[[1,2,3,5],[4],[6]] => [4,2] => [1,2,1,1,1] => 2
[[1,2,3,4],[5],[6]] => [6] => [1,1,1,1,1,1] => 0
[[1,3,5],[2,4,6]] => [2,2,2] => [1,2,2,1] => 2
[[1,2,5],[3,4,6]] => [3,3] => [1,1,2,1,1] => 2
[[1,3,4],[2,5,6]] => [2,3,1] => [2,1,2,1] => 3
[[1,2,4],[3,5,6]] => [3,2,1] => [2,2,1,1] => 1
[[1,2,3],[4,5,6]] => [4,2] => [1,2,1,1,1] => 2
[[1,4,6],[2,5],[3]] => [3,2,1] => [2,2,1,1] => 1
[[1,3,6],[2,5],[4]] => [2,2,2] => [1,2,2,1] => 2
[[1,2,6],[3,5],[4]] => [4,2] => [1,2,1,1,1] => 2
[[1,3,6],[2,4],[5]] => [2,3,1] => [2,1,2,1] => 3
[[1,2,6],[3,4],[5]] => [3,2,1] => [2,2,1,1] => 1
[[1,4,5],[2,6],[3]] => [3,3] => [1,1,2,1,1] => 2
[[1,3,5],[2,6],[4]] => [2,2,2] => [1,2,2,1] => 2
[[1,2,5],[3,6],[4]] => [4,2] => [1,2,1,1,1] => 2
[[1,3,4],[2,6],[5]] => [2,3,1] => [2,1,2,1] => 3
[[1,2,4],[3,6],[5]] => [3,2,1] => [2,2,1,1] => 1
[[1,2,3],[4,6],[5]] => [5,1] => [2,1,1,1,1] => 1
[[1,3,5],[2,4],[6]] => [2,2,2] => [1,2,2,1] => 2
[[1,2,5],[3,4],[6]] => [3,3] => [1,1,2,1,1] => 2
[[1,3,4],[2,5],[6]] => [2,4] => [1,1,1,2,1] => 2
[[1,2,4],[3,5],[6]] => [3,3] => [1,1,2,1,1] => 2
[[1,2,3],[4,5],[6]] => [4,2] => [1,2,1,1,1] => 2
[[1,5,6],[2],[3],[4]] => [4,2] => [1,2,1,1,1] => 2
[[1,4,6],[2],[3],[5]] => [3,2,1] => [2,2,1,1] => 1
[[1,3,6],[2],[4],[5]] => [2,3,1] => [2,1,2,1] => 3
[[1,2,6],[3],[4],[5]] => [5,1] => [2,1,1,1,1] => 1
[[1,4,5],[2],[3],[6]] => [3,3] => [1,1,2,1,1] => 2
[[1,3,5],[2],[4],[6]] => [2,2,2] => [1,2,2,1] => 2
[[1,2,5],[3],[4],[6]] => [4,2] => [1,2,1,1,1] => 2
[[1,3,4],[2],[5],[6]] => [2,4] => [1,1,1,2,1] => 2
[[1,2,4],[3],[5],[6]] => [3,3] => [1,1,2,1,1] => 2
[[1,2,3],[4],[5],[6]] => [6] => [1,1,1,1,1,1] => 0
[[1,4],[2,5],[3,6]] => [3,3] => [1,1,2,1,1] => 2
[[1,3],[2,5],[4,6]] => [2,2,2] => [1,2,2,1] => 2
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Description
The absolute variation of a composition.
Map
conjugate
Description
Map
valley composition
Description
The composition corresponding to the valley 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 valley if $i-1$ is a descent and $i$ is an ascent.
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 valley set of $T$.
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 valley if $i-1$ is a descent and $i$ is an ascent.
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 valley set of $T$.
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