Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00189: Skew partitions —rotate⟶ Skew partitions
St001487: Skew partitions ⟶ ℤ
Values
[1,0,1,0] => [1] => [[1],[]] => [[1],[]] => 1
[1,0,1,0,1,0] => [2,1] => [[2,1],[]] => [[2,2],[1]] => 2
[1,0,1,1,0,0] => [1,1] => [[1,1],[]] => [[1,1],[]] => 1
[1,1,0,0,1,0] => [2] => [[2],[]] => [[2],[]] => 1
[1,1,0,1,0,0] => [1] => [[1],[]] => [[1],[]] => 1
[1,0,1,0,1,1,0,0] => [2,2,1] => [[2,2,1],[]] => [[2,2,2],[1]] => 2
[1,0,1,1,0,0,1,0] => [3,1,1] => [[3,1,1],[]] => [[3,3,3],[2,2]] => 2
[1,0,1,1,0,1,0,0] => [2,1,1] => [[2,1,1],[]] => [[2,2,2],[1,1]] => 2
[1,0,1,1,1,0,0,0] => [1,1,1] => [[1,1,1],[]] => [[1,1,1],[]] => 1
[1,1,0,0,1,0,1,0] => [3,2] => [[3,2],[]] => [[3,3],[1]] => 2
[1,1,0,0,1,1,0,0] => [2,2] => [[2,2],[]] => [[2,2],[]] => 1
[1,1,0,1,0,0,1,0] => [3,1] => [[3,1],[]] => [[3,3],[2]] => 2
[1,1,0,1,0,1,0,0] => [2,1] => [[2,1],[]] => [[2,2],[1]] => 2
[1,1,0,1,1,0,0,0] => [1,1] => [[1,1],[]] => [[1,1],[]] => 1
[1,1,1,0,0,0,1,0] => [3] => [[3],[]] => [[3],[]] => 1
[1,1,1,0,0,1,0,0] => [2] => [[2],[]] => [[2],[]] => 1
[1,1,1,0,1,0,0,0] => [1] => [[1],[]] => [[1],[]] => 1
[1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => [[2,1,1,1],[]] => [[2,2,2,2],[1,1,1]] => 2
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => [[1,1,1,1],[]] => [[1,1,1,1],[]] => 1
[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [[2,2,1],[]] => [[2,2,2],[1]] => 2
[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => [[3,1,1],[]] => [[3,3,3],[2,2]] => 2
[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => [[2,1,1],[]] => [[2,2,2],[1,1]] => 2
[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [[1,1,1],[]] => [[1,1,1],[]] => 1
[1,1,1,0,0,1,0,1,0,0] => [3,2] => [[3,2],[]] => [[3,3],[1]] => 2
[1,1,1,0,0,1,1,0,0,0] => [2,2] => [[2,2],[]] => [[2,2],[]] => 1
[1,1,1,0,1,0,0,0,1,0] => [4,1] => [[4,1],[]] => [[4,4],[3]] => 2
[1,1,1,0,1,0,0,1,0,0] => [3,1] => [[3,1],[]] => [[3,3],[2]] => 2
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [[2,1],[]] => [[2,2],[1]] => 2
[1,1,1,0,1,1,0,0,0,0] => [1,1] => [[1,1],[]] => [[1,1],[]] => 1
[1,1,1,1,0,0,0,0,1,0] => [4] => [[4],[]] => [[4],[]] => 1
[1,1,1,1,0,0,0,1,0,0] => [3] => [[3],[]] => [[3],[]] => 1
[1,1,1,1,0,0,1,0,0,0] => [2] => [[2],[]] => [[2],[]] => 1
[1,1,1,1,0,1,0,0,0,0] => [1] => [[1],[]] => [[1],[]] => 1
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => [[1,1,1,1,1],[]] => [[1,1,1,1,1],[]] => 1
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,1,1,1] => [[2,1,1,1],[]] => [[2,2,2,2],[1,1,1]] => 2
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1] => [[1,1,1,1],[]] => [[1,1,1,1],[]] => 1
[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => [[2,2,1],[]] => [[2,2,2],[1]] => 2
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,1,1] => [[3,1,1],[]] => [[3,3,3],[2,2]] => 2
[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => [[2,1,1],[]] => [[2,2,2],[1,1]] => 2
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [[1,1,1],[]] => [[1,1,1],[]] => 1
[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => [[3,2],[]] => [[3,3],[1]] => 2
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [[2,2],[]] => [[2,2],[]] => 1
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,1] => [[4,1],[]] => [[4,4],[3]] => 2
[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [[3,1],[]] => [[3,3],[2]] => 2
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [[2,1],[]] => [[2,2],[1]] => 2
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [[1,1],[]] => [[1,1],[]] => 1
[1,1,1,1,1,0,0,0,0,0,1,0] => [5] => [[5],[]] => [[5],[]] => 1
[1,1,1,1,1,0,0,0,0,1,0,0] => [4] => [[4],[]] => [[4],[]] => 1
[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => [[3],[]] => [[3],[]] => 1
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [[2],[]] => [[2],[]] => 1
[1,1,1,1,1,0,1,0,0,0,0,0] => [1] => [[1],[]] => [[1],[]] => 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1] => [[1,1,1,1,1],[]] => [[1,1,1,1,1],[]] => 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1] => [[2,1,1,1],[]] => [[2,2,2,2],[1,1,1]] => 2
[1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1] => [[1,1,1,1],[]] => [[1,1,1,1],[]] => 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [2,2,1] => [[2,2,1],[]] => [[2,2,2],[1]] => 2
[1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [3,1,1] => [[3,1,1],[]] => [[3,3,3],[2,2]] => 2
[1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [2,1,1] => [[2,1,1],[]] => [[2,2,2],[1,1]] => 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [1,1,1] => [[1,1,1],[]] => [[1,1,1],[]] => 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [3,2] => [[3,2],[]] => [[3,3],[1]] => 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [2,2] => [[2,2],[]] => [[2,2],[]] => 1
[1,1,1,1,1,0,1,0,0,0,1,0,0,0] => [4,1] => [[4,1],[]] => [[4,4],[3]] => 2
[1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [3,1] => [[3,1],[]] => [[3,3],[2]] => 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [2,1] => [[2,1],[]] => [[2,2],[1]] => 2
[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,1] => [[1,1],[]] => [[1,1],[]] => 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [5] => [[5],[]] => [[5],[]] => 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [4] => [[4],[]] => [[4],[]] => 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [3] => [[3],[]] => [[3],[]] => 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [2] => [[2],[]] => [[2],[]] => 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [1] => [[1],[]] => [[1],[]] => 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1] => [[1,1,1,1,1],[]] => [[1,1,1,1,1],[]] => 1
[1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1] => [[2,1,1,1],[]] => [[2,2,2,2],[1,1,1]] => 2
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1] => [[1,1,1,1],[]] => [[1,1,1,1],[]] => 1
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0] => [2,2,1] => [[2,2,1],[]] => [[2,2,2],[1]] => 2
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0] => [3,1,1] => [[3,1,1],[]] => [[3,3,3],[2,2]] => 2
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0] => [2,1,1] => [[2,1,1],[]] => [[2,2,2],[1,1]] => 2
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0] => [1,1,1] => [[1,1,1],[]] => [[1,1,1],[]] => 1
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0] => [3,2] => [[3,2],[]] => [[3,3],[1]] => 2
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0] => [2,2] => [[2,2],[]] => [[2,2],[]] => 1
[1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0] => [4,1] => [[4,1],[]] => [[4,4],[3]] => 2
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0] => [3,1] => [[3,1],[]] => [[3,3],[2]] => 2
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [2,1] => [[2,1],[]] => [[2,2],[1]] => 2
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [1,1] => [[1,1],[]] => [[1,1],[]] => 1
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0] => [5] => [[5],[]] => [[5],[]] => 1
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [4] => [[4],[]] => [[4],[]] => 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0] => [3] => [[3],[]] => [[3],[]] => 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [2] => [[2],[]] => [[2],[]] => 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [1] => [[1],[]] => [[1],[]] => 1
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1,1] => [[1,1,1,1,1],[]] => [[1,1,1,1,1],[]] => 1
[1,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1,1] => [[2,1,1,1],[]] => [[2,2,2,2],[1,1,1]] => 2
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1] => [[1,1,1,1],[]] => [[1,1,1,1],[]] => 1
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1] => [[2,2,1],[]] => [[2,2,2],[1]] => 2
[1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0] => [3,1,1] => [[3,1,1],[]] => [[3,3,3],[2,2]] => 2
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1] => [[2,1,1],[]] => [[2,2,2],[1,1]] => 2
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1] => [[1,1,1],[]] => [[1,1,1],[]] => 1
[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0] => [3,2] => [[3,2],[]] => [[3,3],[1]] => 2
[1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0] => [2,2] => [[2,2],[]] => [[2,2],[]] => 1
[1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0] => [4,1] => [[4,1],[]] => [[4,4],[3]] => 2
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0] => [3,1] => [[3,1],[]] => [[3,3],[2]] => 2
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [2,1] => [[2,1],[]] => [[2,2],[1]] => 2
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0] => [1,1] => [[1,1],[]] => [[1,1],[]] => 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0] => [5] => [[5],[]] => [[5],[]] => 1
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Description
The number of inner corners of a skew partition.
Map
rotate
Description
The rotation of a skew partition.
This is the skew partition obtained by rotating the diagram by 180 degrees. Equivalently, given a skew partition $\lambda/\mu$, its rotation $(\lambda/\mu)^\natural$ is the skew partition with cells $\{(a-i, b-j)| (i, j) \in \lambda/\mu\}$, where $b$ and $a$ are the first part and the number of parts of $\lambda$ respectively.
This is the skew partition obtained by rotating the diagram by 180 degrees. Equivalently, given a skew partition $\lambda/\mu$, its rotation $(\lambda/\mu)^\natural$ is the skew partition with cells $\{(a-i, b-j)| (i, j) \in \lambda/\mu\}$, where $b$ and $a$ are the first part and the number of parts of $\lambda$ respectively.
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
to skew partition
Description
The partition regarded as a skew partition.
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