Identifier
-
Mp00012:
Binary trees
—to Dyck path: up step, left tree, down step, right tree⟶
Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001438: 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
[[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => [3] => [[3],[]] => 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
[[[.,.],[.,.]],.] => [1,1,1,0,0,1,0,0] => [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
[[[.,[.,.]],.],[.,.]] => [1,1,1,0,1,0,0,0,1,0] => [4,1] => [[4,1],[]] => 0
[[[[.,.],.],.],[.,.]] => [1,1,1,1,0,0,0,0,1,0] => [4] => [[4],[]] => 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
[[[.,[.,.]],[.,.]],.] => [1,1,1,0,1,0,0,1,0,0] => [3,1] => [[3,1],[]] => 0
[[[[.,.],.],[.,.]],.] => [1,1,1,1,0,0,0,1,0,0] => [3] => [[3],[]] => 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
[[[[.,.],[.,.]],.],.] => [1,1,1,1,0,0,1,0,0,0] => [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
[[[[[.,.],.],.],.],[.,.]] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5] => [[5],[]] => 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
[[[[.,[.,.]],.],[.,.]],.] => [1,1,1,1,0,1,0,0,0,1,0,0] => [4,1] => [[4,1],[]] => 0
[[[[[.,.],.],.],[.,.]],.] => [1,1,1,1,1,0,0,0,0,1,0,0] => [4] => [[4],[]] => 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
[[[[.,[.,.]],[.,.]],.],.] => [1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [[3,1],[]] => 0
[[[[[.,.],.],[.,.]],.],.] => [1,1,1,1,1,0,0,0,1,0,0,0] => [3] => [[3],[]] => 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
[[[[[.,.],[.,.]],.],.],.] => [1,1,1,1,1,0,0,1,0,0,0,0] => [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
[[[[[[.,.],.],.],.],[.,.]],.] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [5] => [[5],[]] => 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
[[[[[.,[.,.]],.],[.,.]],.],.] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0] => [4,1] => [[4,1],[]] => 0
[[[[[[.,.],.],.],[.,.]],.],.] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [4] => [[4],[]] => 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
[[[[[.,[.,.]],[.,.]],.],.],.] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [3,1] => [[3,1],[]] => 0
[[[[[[.,.],.],[.,.]],.],.],.] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [3] => [[3],[]] => 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
[[[[[[.,.],[.,.]],.],.],.],.] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [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
[[[[[[[.,.],.],.],.],[.,.]],.],.] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0] => [5] => [[5],[]] => 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
[[[[[[.,[.,.]],.],[.,.]],.],.],.] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0] => [4,1] => [[4,1],[]] => 0
[[[[[[[.,.],.],.],[.,.]],.],.],.] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [4] => [[4],[]] => 0
[[[[[.,[.,[[.,.],.]]],.],.],.],.] => [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0] => [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
[[[[[[.,[.,.]],[.,.]],.],.],.],.] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0] => [3,1] => [[3,1],[]] => 0
[[[[[[[.,.],.],[.,.]],.],.],.],.] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0] => [3] => [[3],[]] => 0
[[[[[[.,[.,[.,.]]],.],.],.],.],.] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [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
[[[[[[[.,.],[.,.]],.],.],.],.],.] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [2] => [[2],[]] => 0
[[[[[[[.,[.,.]],.],.],.],.],.],.] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [1] => [[1],[]] => 0
[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.] => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [2,1] => [[2,1],[]] => 0
[[[[[[[[.,[.,.]],.],.],.],.],.],.],.] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [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
[[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.],.] => [1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0] => [2,1] => [[2,1],[]] => 0
[[[[[[[[.,.],[.,.]],.],.],.],.],.],.] => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0] => [2] => [[2],[]] => 0
[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.] => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0] => [1] => [[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
[[[[[[[.,[[.,.],.]],.],.],.],.],.],.] => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0] => [1,1] => [[1,1],[]] => 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
[[[[[[[[.,.],.],[.,.]],.],.],.],.],.] => [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0] => [3] => [[3],[]] => 0
[[[[[[[[.,.],.],.],[.,.]],.],.],.],.] => [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0] => [4] => [[4],[]] => 0
[[[[[[[[.,.],.],.],.],[.,.]],.],.],.] => [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0] => [5] => [[5],[]] => 0
[[[[[[[[[.,.],[.,.]],.],.],.],.],.],.],.] => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0] => [2] => [[2],[]] => 0
[[[[[[[[[.,.],.],[.,.]],.],.],.],.],.],.] => [1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0] => [3] => [[3],[]] => 0
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search for individual values
searching the database for the individual values of this statistic
Description
The number of missing boxes of a skew partition.
Map
to Dyck path: up step, left tree, down step, right tree
Description
Return the associated Dyck path, using the bijection 1L0R.
This is given recursively as follows:
This is given recursively as follows:
- a leaf is associated to the empty Dyck Word
- a tree with children $l,r$ is associated with the Dyck path described by 1L0R where $L$ and $R$ are respectively the Dyck words associated with the trees $l$ and $r$.
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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