Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤ
Values
[1,0,1,0] => [1] => [[1],[]] => 0
[1,0,1,0,1,0] => [2,1] => [[2,1],[]] => 0
[1,0,1,1,0,0] => [1,1] => [[1,1],[]] => 0
[1,1,0,0,1,0] => [2] => [[2],[]] => 0
[1,1,0,1,0,0] => [1] => [[1],[]] => 0
[1,0,1,0,1,1,0,0] => [2,2,1] => [[2,2,1],[]] => 0
[1,0,1,1,0,0,1,0] => [3,1,1] => [[3,1,1],[]] => 0
[1,0,1,1,0,1,0,0] => [2,1,1] => [[2,1,1],[]] => 0
[1,0,1,1,1,0,0,0] => [1,1,1] => [[1,1,1],[]] => 0
[1,1,0,0,1,0,1,0] => [3,2] => [[3,2],[]] => 0
[1,1,0,0,1,1,0,0] => [2,2] => [[2,2],[]] => 0
[1,1,0,1,0,0,1,0] => [3,1] => [[3,1],[]] => 0
[1,1,0,1,0,1,0,0] => [2,1] => [[2,1],[]] => 0
[1,1,0,1,1,0,0,0] => [1,1] => [[1,1],[]] => 0
[1,1,1,0,0,0,1,0] => [3] => [[3],[]] => 0
[1,1,1,0,0,1,0,0] => [2] => [[2],[]] => 0
[1,1,1,0,1,0,0,0] => [1] => [[1],[]] => 0
[1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => [[2,1,1,1],[]] => 0
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => [[1,1,1,1],[]] => 0
[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [[2,2,1],[]] => 0
[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => [[3,1,1],[]] => 0
[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => [[2,1,1],[]] => 0
[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [[1,1,1],[]] => 0
[1,1,1,0,0,1,0,1,0,0] => [3,2] => [[3,2],[]] => 0
[1,1,1,0,0,1,1,0,0,0] => [2,2] => [[2,2],[]] => 0
[1,1,1,0,1,0,0,0,1,0] => [4,1] => [[4,1],[]] => 0
[1,1,1,0,1,0,0,1,0,0] => [3,1] => [[3,1],[]] => 0
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [[2,1],[]] => 0
[1,1,1,0,1,1,0,0,0,0] => [1,1] => [[1,1],[]] => 0
[1,1,1,1,0,0,0,0,1,0] => [4] => [[4],[]] => 0
[1,1,1,1,0,0,0,1,0,0] => [3] => [[3],[]] => 0
[1,1,1,1,0,0,1,0,0,0] => [2] => [[2],[]] => 0
[1,1,1,1,0,1,0,0,0,0] => [1] => [[1],[]] => 0
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 0
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,1,1,1] => [[2,1,1,1],[]] => 0
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1] => [[1,1,1,1],[]] => 0
[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => [[2,2,1],[]] => 0
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,1,1] => [[3,1,1],[]] => 0
[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => [[2,1,1],[]] => 0
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [[1,1,1],[]] => 0
[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => [[3,2],[]] => 0
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [[2,2],[]] => 0
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,1] => [[4,1],[]] => 0
[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [[3,1],[]] => 0
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [[2,1],[]] => 0
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [[1,1],[]] => 0
[1,1,1,1,1,0,0,0,0,0,1,0] => [5] => [[5],[]] => 0
[1,1,1,1,1,0,0,0,0,1,0,0] => [4] => [[4],[]] => 0
[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => [[3],[]] => 0
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [[2],[]] => 0
[1,1,1,1,1,0,1,0,0,0,0,0] => [1] => [[1],[]] => 0
[1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 0
[1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1] => [[2,1,1,1],[]] => 0
[1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1] => [[1,1,1,1],[]] => 0
[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [2,2,1] => [[2,2,1],[]] => 0
[1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [3,1,1] => [[3,1,1],[]] => 0
[1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [2,1,1] => [[2,1,1],[]] => 0
[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [1,1,1] => [[1,1,1],[]] => 0
[1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [3,2] => [[3,2],[]] => 0
[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [2,2] => [[2,2],[]] => 0
[1,1,1,1,1,0,1,0,0,0,1,0,0,0] => [4,1] => [[4,1],[]] => 0
[1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [3,1] => [[3,1],[]] => 0
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [2,1] => [[2,1],[]] => 0
[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,1] => [[1,1],[]] => 0
[1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [5] => [[5],[]] => 0
[1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [4] => [[4],[]] => 0
[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [3] => [[3],[]] => 0
[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [2] => [[2],[]] => 0
[1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [1] => [[1],[]] => 0
[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],[]] => 0
[1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1] => [[2,1,1,1],[]] => 0
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1] => [[1,1,1,1],[]] => 0
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0] => [2,2,1] => [[2,2,1],[]] => 0
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0] => [3,1,1] => [[3,1,1],[]] => 0
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0] => [2,1,1] => [[2,1,1],[]] => 0
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0] => [1,1,1] => [[1,1,1],[]] => 0
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0] => [3,2] => [[3,2],[]] => 0
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0] => [2,2] => [[2,2],[]] => 0
[1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0] => [4,1] => [[4,1],[]] => 0
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0] => [3,1] => [[3,1],[]] => 0
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [2,1] => [[2,1],[]] => 0
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [1,1] => [[1,1],[]] => 0
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0] => [5] => [[5],[]] => 0
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [4] => [[4],[]] => 0
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0] => [3] => [[3],[]] => 0
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [2] => [[2],[]] => 0
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [1] => [[1],[]] => 0
[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],[]] => 0
[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],[]] => 0
[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],[]] => 0
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1] => [[2,2,1],[]] => 0
[1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0] => [3,1,1] => [[3,1,1],[]] => 0
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1] => [[2,1,1],[]] => 0
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1] => [[1,1,1],[]] => 0
[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0] => [3,2] => [[3,2],[]] => 0
[1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0] => [2,2] => [[2,2],[]] => 0
[1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0] => [4,1] => [[4,1],[]] => 0
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0] => [3,1] => [[3,1],[]] => 0
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [2,1] => [[2,1],[]] => 0
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0] => [1,1] => [[1,1],[]] => 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0] => [5] => [[5],[]] => 0
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Description
The number of missing boxes in the first row.
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 λ associated to a Dyck path is defined to be the complementary partition inside the staircase partition (n−1,…,2,1) when cutting out D considered as a path from (0,0) to (n,n).
In other words, λi 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,…,2,1).
The partition λ associated to a Dyck path is defined to be the complementary partition inside the staircase partition (n−1,…,2,1) when cutting out D considered as a path from (0,0) to (n,n).
In other words, λi 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,…,2,1).
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