Identifier
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Mp00207:
Standard tableaux
—horizontal strip sizes⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001514: Dyck paths ⟶ ℤ
Values
[[1]] => [1] => [1,0] => [1,0] => 1
[[1,2]] => [2] => [1,1,0,0] => [1,0,1,0] => 1
[[1],[2]] => [1,1] => [1,0,1,0] => [1,1,0,0] => 1
[[1,2,3]] => [3] => [1,1,1,0,0,0] => [1,1,0,1,0,0] => 1
[[1,3],[2]] => [1,2] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 2
[[1,2],[3]] => [2,1] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => 2
[[1],[2],[3]] => [1,1,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 1
[[1,2,3,4]] => [4] => [1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0] => 1
[[1,3,4],[2]] => [1,3] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => 2
[[1,2,4],[3]] => [2,2] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => 3
[[1,2,3],[4]] => [3,1] => [1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => 2
[[1,3],[2,4]] => [1,2,1] => [1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 3
[[1,2],[3,4]] => [2,2] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => 3
[[1,4],[2],[3]] => [1,1,2] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 2
[[1,3],[2],[4]] => [1,2,1] => [1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 3
[[1,2],[3],[4]] => [2,1,1] => [1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => 2
[[1],[2],[3],[4]] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 1
[[1,2,3,4,5]] => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 1
[[1,3,4,5],[2]] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => 2
[[1,2,4,5],[3]] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,1,0,0] => 3
[[1,2,3,5],[4]] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 3
[[1,2,3,4],[5]] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 2
[[1,3,5],[2,4]] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 4
[[1,2,5],[3,4]] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,1,0,0] => 3
[[1,3,4],[2,5]] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => 3
[[1,2,4],[3,5]] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => 4
[[1,2,3],[4,5]] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 3
[[1,4,5],[2],[3]] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[[1,3,5],[2],[4]] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 4
[[1,2,5],[3],[4]] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 3
[[1,3,4],[2],[5]] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => 3
[[1,2,4],[3],[5]] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => 4
[[1,2,3],[4],[5]] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => 2
[[1,4],[2,5],[3]] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 3
[[1,3],[2,5],[4]] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 4
[[1,2],[3,5],[4]] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 3
[[1,3],[2,4],[5]] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 3
[[1,2],[3,4],[5]] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => 4
[[1,5],[2],[3],[4]] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 2
[[1,4],[2],[3],[5]] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 3
[[1,3],[2],[4],[5]] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 3
[[1,2],[3],[4],[5]] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => 2
[[1],[2],[3],[4],[5]] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1
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Description
The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule.
Map
horizontal strip sizes
Description
The composition of horizontal strip sizes.
We associate to a standard Young tableau $T$ the composition $(c_1,\dots,c_k)$, such that $k$ is minimal and the numbers $c_1+\dots+c_i + 1,\dots,c_1+\dots+c_{i+1}$ form a horizontal strip in $T$ for all $i$.
We associate to a standard Young tableau $T$ the composition $(c_1,\dots,c_k)$, such that $k$ is minimal and the numbers $c_1+\dots+c_i + 1,\dots,c_1+\dots+c_{i+1}$ form a horizontal strip in $T$ for all $i$.
Map
peaks-to-valleys
Description
Return the path that has a valley wherever the original path has a peak of height at least one.
More precisely, the height of a valley in the image is the height of the corresponding peak minus $2$.
This is also (the inverse of) rowmotion on Dyck paths regarded as order ideals in the triangular poset.
More precisely, the height of a valley in the image is the height of the corresponding peak minus $2$.
This is also (the inverse of) rowmotion on Dyck paths regarded as order ideals in the triangular poset.
Map
bounce path
Description
The bounce path determined by an integer composition.
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