Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000091: Integer compositions ⟶ ℤ
Values
[1,0,1,0] => [1] => [[1]] => [1] => 0
[1,0,1,0,1,0] => [2,1] => [[1,2],[3]] => [2,1] => 0
[1,0,1,1,0,0] => [1,1] => [[1],[2]] => [1,1] => 0
[1,1,0,0,1,0] => [2] => [[1,2]] => [2] => 0
[1,1,0,1,0,0] => [1] => [[1]] => [1] => 0
[1,0,1,0,1,0,1,0] => [3,2,1] => [[1,2,3],[4,5],[6]] => [3,2,1] => 0
[1,0,1,0,1,1,0,0] => [2,2,1] => [[1,2],[3,4],[5]] => [2,2,1] => 0
[1,0,1,1,0,0,1,0] => [3,1,1] => [[1,2,3],[4],[5]] => [3,1,1] => 0
[1,0,1,1,0,1,0,0] => [2,1,1] => [[1,2],[3],[4]] => [2,1,1] => 0
[1,0,1,1,1,0,0,0] => [1,1,1] => [[1],[2],[3]] => [1,1,1] => 0
[1,1,0,0,1,0,1,0] => [3,2] => [[1,2,3],[4,5]] => [3,2] => 0
[1,1,0,0,1,1,0,0] => [2,2] => [[1,2],[3,4]] => [2,2] => 0
[1,1,0,1,0,0,1,0] => [3,1] => [[1,2,3],[4]] => [3,1] => 0
[1,1,0,1,0,1,0,0] => [2,1] => [[1,2],[3]] => [2,1] => 0
[1,1,0,1,1,0,0,0] => [1,1] => [[1],[2]] => [1,1] => 0
[1,1,1,0,0,0,1,0] => [3] => [[1,2,3]] => [3] => 0
[1,1,1,0,0,1,0,0] => [2] => [[1,2]] => [2] => 0
[1,1,1,0,1,0,0,0] => [1] => [[1]] => [1] => 0
[1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [2,2,2,1] => 0
[1,0,1,1,0,1,0,1,0,0] => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => [3,2,1,1] => 0
[1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => [[1,2],[3,4],[5],[6]] => [2,2,1,1] => 0
[1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [4,1,1,1] => 0
[1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => [3,1,1,1] => 0
[1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => [[1,2],[3],[4],[5]] => [2,1,1,1] => 0
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => [[1],[2],[3],[4]] => [1,1,1,1] => 0
[1,1,0,0,1,1,0,1,0,0] => [3,2,2] => [[1,2,3],[4,5],[6,7]] => [3,2,2] => 0
[1,1,0,0,1,1,1,0,0,0] => [2,2,2] => [[1,2],[3,4],[5,6]] => [2,2,2] => 0
[1,1,0,1,0,0,1,1,0,0] => [3,3,1] => [[1,2,3],[4,5,6],[7]] => [3,3,1] => 0
[1,1,0,1,0,1,0,0,1,0] => [4,2,1] => [[1,2,3,4],[5,6],[7]] => [4,2,1] => 0
[1,1,0,1,0,1,0,1,0,0] => [3,2,1] => [[1,2,3],[4,5],[6]] => [3,2,1] => 0
[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [[1,2],[3,4],[5]] => [2,2,1] => 0
[1,1,0,1,1,0,0,0,1,0] => [4,1,1] => [[1,2,3,4],[5],[6]] => [4,1,1] => 0
[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => [[1,2,3],[4],[5]] => [3,1,1] => 0
[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => [[1,2],[3],[4]] => [2,1,1] => 0
[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [[1],[2],[3]] => [1,1,1] => 0
[1,1,1,0,0,0,1,0,1,0] => [4,3] => [[1,2,3,4],[5,6,7]] => [4,3] => 0
[1,1,1,0,0,0,1,1,0,0] => [3,3] => [[1,2,3],[4,5,6]] => [3,3] => 0
[1,1,1,0,0,1,0,0,1,0] => [4,2] => [[1,2,3,4],[5,6]] => [4,2] => 0
[1,1,1,0,0,1,0,1,0,0] => [3,2] => [[1,2,3],[4,5]] => [3,2] => 0
[1,1,1,0,0,1,1,0,0,0] => [2,2] => [[1,2],[3,4]] => [2,2] => 0
[1,1,1,0,1,0,0,0,1,0] => [4,1] => [[1,2,3,4],[5]] => [4,1] => 0
[1,1,1,0,1,0,0,1,0,0] => [3,1] => [[1,2,3],[4]] => [3,1] => 0
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [[1,2],[3]] => [2,1] => 0
[1,1,1,0,1,1,0,0,0,0] => [1,1] => [[1],[2]] => [1,1] => 0
[1,1,1,1,0,0,0,0,1,0] => [4] => [[1,2,3,4]] => [4] => 0
[1,1,1,1,0,0,0,1,0,0] => [3] => [[1,2,3]] => [3] => 0
[1,1,1,1,0,0,1,0,0,0] => [2] => [[1,2]] => [2] => 0
[1,1,1,1,0,1,0,0,0,0] => [1] => [[1]] => [1] => 0
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [2,2,1,1,1] => 0
[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [3,1,1,1,1] => 0
[1,0,1,1,1,1,0,1,0,0,0,0] => [2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [2,1,1,1,1] => 0
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [1,1,1,1,1] => 0
[1,1,0,1,0,1,1,1,0,0,0,0] => [2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [2,2,2,1] => 0
[1,1,0,1,1,0,1,0,1,0,0,0] => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => [3,2,1,1] => 0
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,2,1,1] => [[1,2],[3,4],[5],[6]] => [2,2,1,1] => 0
[1,1,0,1,1,1,0,0,0,1,0,0] => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [4,1,1,1] => 0
[1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => [3,1,1,1] => 0
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,1,1,1] => [[1,2],[3],[4],[5]] => [2,1,1,1] => 0
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1] => [[1],[2],[3],[4]] => [1,1,1,1] => 0
[1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,2] => [[1,2,3],[4,5],[6,7]] => [3,2,2] => 0
[1,1,1,0,0,1,1,1,0,0,0,0] => [2,2,2] => [[1,2],[3,4],[5,6]] => [2,2,2] => 0
[1,1,1,0,1,0,0,1,1,0,0,0] => [3,3,1] => [[1,2,3],[4,5,6],[7]] => [3,3,1] => 0
[1,1,1,0,1,0,1,0,0,1,0,0] => [4,2,1] => [[1,2,3,4],[5,6],[7]] => [4,2,1] => 0
[1,1,1,0,1,0,1,0,1,0,0,0] => [3,2,1] => [[1,2,3],[4,5],[6]] => [3,2,1] => 0
[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => [[1,2],[3,4],[5]] => [2,2,1] => 0
[1,1,1,0,1,1,0,0,0,0,1,0] => [5,1,1] => [[1,2,3,4,5],[6],[7]] => [5,1,1] => 0
[1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,1] => [[1,2,3,4],[5],[6]] => [4,1,1] => 0
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,1,1] => [[1,2,3],[4],[5]] => [3,1,1] => 0
[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => [[1,2],[3],[4]] => [2,1,1] => 0
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [[1],[2],[3]] => [1,1,1] => 0
[1,1,1,1,0,0,0,1,0,1,0,0] => [4,3] => [[1,2,3,4],[5,6,7]] => [4,3] => 0
[1,1,1,1,0,0,0,1,1,0,0,0] => [3,3] => [[1,2,3],[4,5,6]] => [3,3] => 0
[1,1,1,1,0,0,1,0,0,0,1,0] => [5,2] => [[1,2,3,4,5],[6,7]] => [5,2] => 0
[1,1,1,1,0,0,1,0,0,1,0,0] => [4,2] => [[1,2,3,4],[5,6]] => [4,2] => 0
[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => [[1,2,3],[4,5]] => [3,2] => 0
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [[1,2],[3,4]] => [2,2] => 0
[1,1,1,1,0,1,0,0,0,0,1,0] => [5,1] => [[1,2,3,4,5],[6]] => [5,1] => 0
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,1] => [[1,2,3,4],[5]] => [4,1] => 0
[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [[1,2,3],[4]] => [3,1] => 0
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [[1,2],[3]] => [2,1] => 0
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [[1],[2]] => [1,1] => 0
[1,1,1,1,1,0,0,0,0,0,1,0] => [5] => [[1,2,3,4,5]] => [5] => 0
[1,1,1,1,1,0,0,0,0,1,0,0] => [4] => [[1,2,3,4]] => [4] => 0
[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => [[1,2,3]] => [3] => 0
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [[1,2]] => [2] => 0
[1,1,1,1,1,0,1,0,0,0,0,0] => [1] => [[1]] => [1] => 0
[1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [2,1,1,1,1,1] => 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [1,1,1,1,1,1] => 0
[1,1,0,1,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [2,2,1,1,1] => 0
[1,1,0,1,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [3,1,1,1,1] => 0
[1,1,0,1,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [2,1,1,1,1] => 0
[1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [1,1,1,1,1] => 0
[1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [2,2,2,1] => 0
[1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => [3,2,1,1] => 0
[1,1,1,0,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1] => [[1,2],[3,4],[5],[6]] => [2,2,1,1] => 0
[1,1,1,0,1,1,1,0,0,0,1,0,0,0] => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [4,1,1,1] => 0
[1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => [3,1,1,1] => 0
[1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1] => [[1,2],[3],[4],[5]] => [2,1,1,1] => 0
[1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1] => [[1],[2],[3],[4]] => [1,1,1,1] => 0
[1,1,1,1,0,0,1,1,0,1,0,0,0,0] => [3,2,2] => [[1,2,3],[4,5],[6,7]] => [3,2,2] => 0
[1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [2,2,2] => [[1,2],[3,4],[5,6]] => [2,2,2] => 0
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Description
The descent variation of a composition.
Defined in [1].
Defined in [1].
Map
to partition
Description
The cut-out partition of a Dyck path.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
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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