Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001595: Skew partitions ⟶ ℤ
Values
[1,0,1,0] => [1] => [[1],[]] => 1
[1,0,1,0,1,0] => [2,1] => [[2,1],[]] => 2
[1,0,1,1,0,0] => [1,1] => [[1,1],[]] => 1
[1,1,0,0,1,0] => [2] => [[2],[]] => 1
[1,1,0,1,0,0] => [1] => [[1],[]] => 1
[1,0,1,0,1,0,1,0] => [3,2,1] => [[3,2,1],[]] => 16
[1,0,1,0,1,1,0,0] => [2,2,1] => [[2,2,1],[]] => 5
[1,0,1,1,0,0,1,0] => [3,1,1] => [[3,1,1],[]] => 6
[1,0,1,1,0,1,0,0] => [2,1,1] => [[2,1,1],[]] => 3
[1,0,1,1,1,0,0,0] => [1,1,1] => [[1,1,1],[]] => 1
[1,1,0,0,1,0,1,0] => [3,2] => [[3,2],[]] => 5
[1,1,0,0,1,1,0,0] => [2,2] => [[2,2],[]] => 2
[1,1,0,1,0,0,1,0] => [3,1] => [[3,1],[]] => 3
[1,1,0,1,0,1,0,0] => [2,1] => [[2,1],[]] => 2
[1,1,0,1,1,0,0,0] => [1,1] => [[1,1],[]] => 1
[1,1,1,0,0,0,1,0] => [3] => [[3],[]] => 1
[1,1,1,0,0,1,0,0] => [2] => [[2],[]] => 1
[1,1,1,0,1,0,0,0] => [1] => [[1],[]] => 1
[1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => [[2,2,2,1],[]] => 14
[1,0,1,1,0,1,0,1,0,0] => [3,2,1,1] => [[3,2,1,1],[]] => 35
[1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => [[2,2,1,1],[]] => 9
[1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => [[4,1,1,1],[]] => 20
[1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => [[3,1,1,1],[]] => 10
[1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => [[2,1,1,1],[]] => 4
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => [[1,1,1,1],[]] => 1
[1,1,0,0,1,1,0,1,0,0] => [3,2,2] => [[3,2,2],[]] => 21
[1,1,0,0,1,1,1,0,0,0] => [2,2,2] => [[2,2,2],[]] => 5
[1,1,0,1,0,0,1,1,0,0] => [3,3,1] => [[3,3,1],[]] => 21
[1,1,0,1,0,1,0,0,1,0] => [4,2,1] => [[4,2,1],[]] => 35
[1,1,0,1,0,1,0,1,0,0] => [3,2,1] => [[3,2,1],[]] => 16
[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [[2,2,1],[]] => 5
[1,1,0,1,1,0,0,0,1,0] => [4,1,1] => [[4,1,1],[]] => 10
[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => [[3,1,1],[]] => 6
[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => [[2,1,1],[]] => 3
[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [[1,1,1],[]] => 1
[1,1,1,0,0,0,1,0,1,0] => [4,3] => [[4,3],[]] => 14
[1,1,1,0,0,0,1,1,0,0] => [3,3] => [[3,3],[]] => 5
[1,1,1,0,0,1,0,0,1,0] => [4,2] => [[4,2],[]] => 9
[1,1,1,0,0,1,0,1,0,0] => [3,2] => [[3,2],[]] => 5
[1,1,1,0,0,1,1,0,0,0] => [2,2] => [[2,2],[]] => 2
[1,1,1,0,1,0,0,0,1,0] => [4,1] => [[4,1],[]] => 4
[1,1,1,0,1,0,0,1,0,0] => [3,1] => [[3,1],[]] => 3
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [[2,1],[]] => 2
[1,1,1,0,1,1,0,0,0,0] => [1,1] => [[1,1],[]] => 1
[1,1,1,1,0,0,0,0,1,0] => [4] => [[4],[]] => 1
[1,1,1,1,0,0,0,1,0,0] => [3] => [[3],[]] => 1
[1,1,1,1,0,0,1,0,0,0] => [2] => [[2],[]] => 1
[1,1,1,1,0,1,0,0,0,0] => [1] => [[1],[]] => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,2,1,1,1] => [[2,2,1,1,1],[]] => 14
[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,1,1,1] => [[3,1,1,1,1],[]] => 15
[1,0,1,1,1,1,0,1,0,0,0,0] => [2,1,1,1,1] => [[2,1,1,1,1],[]] => 5
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 1
[1,1,0,1,0,1,1,1,0,0,0,0] => [2,2,2,1] => [[2,2,2,1],[]] => 14
[1,1,0,1,1,0,1,0,1,0,0,0] => [3,2,1,1] => [[3,2,1,1],[]] => 35
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,2,1,1] => [[2,2,1,1],[]] => 9
[1,1,0,1,1,1,0,0,0,1,0,0] => [4,1,1,1] => [[4,1,1,1],[]] => 20
[1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,1,1] => [[3,1,1,1],[]] => 10
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,1,1,1] => [[2,1,1,1],[]] => 4
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1] => [[1,1,1,1],[]] => 1
[1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,2] => [[3,2,2],[]] => 21
[1,1,1,0,0,1,1,1,0,0,0,0] => [2,2,2] => [[2,2,2],[]] => 5
[1,1,1,0,1,0,0,1,1,0,0,0] => [3,3,1] => [[3,3,1],[]] => 21
[1,1,1,0,1,0,1,0,0,1,0,0] => [4,2,1] => [[4,2,1],[]] => 35
[1,1,1,0,1,0,1,0,1,0,0,0] => [3,2,1] => [[3,2,1],[]] => 16
[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => [[2,2,1],[]] => 5
[1,1,1,0,1,1,0,0,0,0,1,0] => [5,1,1] => [[5,1,1],[]] => 15
[1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,1] => [[4,1,1],[]] => 10
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,1,1] => [[3,1,1],[]] => 6
[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => [[2,1,1],[]] => 3
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [[1,1,1],[]] => 1
[1,1,1,1,0,0,0,1,0,1,0,0] => [4,3] => [[4,3],[]] => 14
[1,1,1,1,0,0,0,1,1,0,0,0] => [3,3] => [[3,3],[]] => 5
[1,1,1,1,0,0,1,0,0,0,1,0] => [5,2] => [[5,2],[]] => 14
[1,1,1,1,0,0,1,0,0,1,0,0] => [4,2] => [[4,2],[]] => 9
[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => [[3,2],[]] => 5
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [[2,2],[]] => 2
[1,1,1,1,0,1,0,0,0,0,1,0] => [5,1] => [[5,1],[]] => 5
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,1] => [[4,1],[]] => 4
[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [[3,1],[]] => 3
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [[2,1],[]] => 2
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [[1,1],[]] => 1
[1,1,1,1,1,0,0,0,0,0,1,0] => [5] => [[5],[]] => 1
[1,1,1,1,1,0,0,0,0,1,0,0] => [4] => [[4],[]] => 1
[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => [[3],[]] => 1
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [[2],[]] => 1
[1,1,1,1,1,0,1,0,0,0,0,0] => [1] => [[1],[]] => 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1,1,1] => [[2,1,1,1,1,1],[]] => 6
[1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => 1
[1,1,0,1,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1,1] => [[2,2,1,1,1],[]] => 14
[1,1,0,1,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1,1] => [[3,1,1,1,1],[]] => 15
[1,1,0,1,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1,1] => [[2,1,1,1,1],[]] => 5
[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
[1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [2,2,2,1] => [[2,2,2,1],[]] => 14
[1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [3,2,1,1] => [[3,2,1,1],[]] => 35
[1,1,1,0,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1] => [[2,2,1,1],[]] => 9
[1,1,1,0,1,1,1,0,0,0,1,0,0,0] => [4,1,1,1] => [[4,1,1,1],[]] => 20
[1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1] => [[3,1,1,1],[]] => 10
[1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1] => [[2,1,1,1],[]] => 4
[1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1] => [[1,1,1,1],[]] => 1
[1,1,1,1,0,0,1,1,0,1,0,0,0,0] => [3,2,2] => [[3,2,2],[]] => 21
[1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [2,2,2] => [[2,2,2],[]] => 5
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Description
The number of standard Young tableaux of the skew partition.
Map
to skew partition
Description
The partition regarded as a skew partition.
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 $(n-1,\ldots,2,1)$.
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