Identifier
-
Mp00207:
Standard tableaux
—horizontal strip sizes⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001232: Dyck paths ⟶ ℤ
Values
[[1]] => [1] => [1,0] => [1,0] => 0
[[1,2]] => [2] => [1,1,0,0] => [1,1,0,0] => 0
[[1],[2]] => [1,1] => [1,0,1,0] => [1,0,1,0] => 1
[[1,2,3]] => [3] => [1,1,1,0,0,0] => [1,1,1,0,0,0] => 0
[[1,3],[2]] => [1,2] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
[[1,2],[3]] => [2,1] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
[[1,2,3,4]] => [4] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => 0
[[1,3,4],[2]] => [1,3] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 3
[[1,2,4],[3]] => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,0,0] => 2
[[1,2,3],[4]] => [3,1] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 1
[[1,2],[3,4]] => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,0,0] => 2
[[1,2,3,4,5]] => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 0
[[1,3,4,5],[2]] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4
[[1,2,4,5],[3]] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => 3
[[1,2,3,5],[4]] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[[1,2,3,4],[5]] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 1
[[1,2,5],[3,4]] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => 3
[[1,2,3],[4,5]] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[[1,2,3,4,5,6]] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
[[1,3,4,5,6],[2]] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 5
[[1,2,4,5,6],[3]] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 4
[[1,2,3,5,6],[4]] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 3
[[1,2,3,4,6],[5]] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
[[1,2,3,4,5],[6]] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[[1,2,5,6],[3,4]] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 4
[[1,2,3,6],[4,5]] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 3
[[1,2,3,4],[5,6]] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
[[1,2,3],[4,5,6]] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 3
[[1,2,3,4,5,6,7]] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 0
[[1,3,4,5,6,7],[2]] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => 6
[[1,2,4,5,6,7],[3]] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 5
[[1,2,3,5,6,7],[4]] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 4
[[1,2,3,4,6,7],[5]] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 3
[[1,2,3,4,5,7],[6]] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 2
[[1,2,3,4,5,6],[7]] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[[1,2,5,6,7],[3,4]] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 5
[[1,2,3,6,7],[4,5]] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 4
[[1,2,3,4,7],[5,6]] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 3
[[1,2,3,4,5],[6,7]] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 2
[[1,2,3,7],[4,5,6]] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 4
[[1,2,3,4],[5,6,7]] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 3
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Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Map
switch returns and last double rise
Description
An alternative to the Adin-Bagno-Roichman transformation of a Dyck path.
This is a bijection preserving the number of up steps before each peak and exchanging the number of components with the position of the last double rise.
This is a bijection preserving the number of up steps before each peak and exchanging the number of components with the position of the last double rise.
Map
bounce path
Description
The bounce path determined by an integer composition.
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 $i$.
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