Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001291: Dyck paths ⟶ ℤ
Values
[1,0,1,0] => [1] => [1] => [1,0] => 1
[1,0,1,0,1,0] => [2,1] => [3] => [1,0,1,0,1,0] => 3
[1,0,1,1,0,0] => [1,1] => [2] => [1,0,1,0] => 2
[1,1,0,0,1,0] => [2] => [1,1] => [1,1,0,0] => 1
[1,1,0,1,0,0] => [1] => [1] => [1,0] => 1
[1,0,1,0,1,0,1,0] => [3,2,1] => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5
[1,0,1,0,1,1,0,0] => [2,2,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => 3
[1,0,1,1,0,0,1,0] => [3,1,1] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 4
[1,0,1,1,0,1,0,0] => [2,1,1] => [2,2] => [1,1,1,0,0,0] => 1
[1,0,1,1,1,0,0,0] => [1,1,1] => [2,1] => [1,0,1,1,0,0] => 2
[1,1,0,0,1,0,1,0] => [3,2] => [5] => [1,0,1,0,1,0,1,0,1,0] => 5
[1,1,0,0,1,1,0,0] => [2,2] => [4] => [1,0,1,0,1,0,1,0] => 4
[1,1,0,1,0,0,1,0] => [3,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 3
[1,1,0,1,0,1,0,0] => [2,1] => [3] => [1,0,1,0,1,0] => 3
[1,1,0,1,1,0,0,0] => [1,1] => [2] => [1,0,1,0] => 2
[1,1,1,0,0,0,1,0] => [3] => [1,1,1] => [1,1,0,1,0,0] => 2
[1,1,1,0,0,1,0,0] => [2] => [1,1] => [1,1,0,0] => 1
[1,1,1,0,1,0,0,0] => [1] => [1] => [1,0] => 1
[1,0,1,0,1,0,1,1,0,0] => [3,3,2,1] => [3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => 3
[1,0,1,0,1,1,0,1,0,0] => [3,2,2,1] => [3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => 5
[1,0,1,1,0,1,0,0,1,0] => [4,2,1,1] => [4,4] => [1,1,1,0,1,0,1,0,0,0] => 3
[1,0,1,1,0,1,0,1,0,0] => [3,2,1,1] => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => 4
[1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 5
[1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 5
[1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => [3,3] => [1,1,1,0,1,0,0,0] => 2
[1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 4
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => [3,1] => [1,0,1,0,1,1,0,0] => 3
[1,1,0,0,1,0,1,1,0,0] => [3,3,2] => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => 2
[1,1,0,0,1,1,0,1,0,0] => [3,2,2] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => 4
[1,1,0,0,1,1,1,0,0,0] => [2,2,2] => [2,2,2] => [1,1,1,1,0,0,0,0] => 1
[1,1,0,1,0,1,0,1,0,0] => [3,2,1] => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5
[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => 3
[1,1,0,1,1,0,0,0,1,0] => [4,1,1] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 5
[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 4
[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => [2,2] => [1,1,1,0,0,0] => 1
[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [2,1] => [1,0,1,1,0,0] => 2
[1,1,1,0,0,0,1,1,0,0] => [3,3] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
[1,1,1,0,0,1,0,0,1,0] => [4,2] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 4
[1,1,1,0,0,1,0,1,0,0] => [3,2] => [5] => [1,0,1,0,1,0,1,0,1,0] => 5
[1,1,1,0,0,1,1,0,0,0] => [2,2] => [4] => [1,0,1,0,1,0,1,0] => 4
[1,1,1,0,1,0,0,0,1,0] => [4,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 4
[1,1,1,0,1,0,0,1,0,0] => [3,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 3
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [3] => [1,0,1,0,1,0] => 3
[1,1,1,0,1,1,0,0,0,0] => [1,1] => [2] => [1,0,1,0] => 2
[1,1,1,1,0,0,0,0,1,0] => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 3
[1,1,1,1,0,0,0,1,0,0] => [3] => [1,1,1] => [1,1,0,1,0,0] => 2
[1,1,1,1,0,0,1,0,0,0] => [2] => [1,1] => [1,1,0,0] => 1
[1,1,1,1,0,1,0,0,0,0] => [1] => [1] => [1,0] => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,2,2,2,1] => [5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => 4
[1,0,1,1,0,1,1,0,0,1,0,0] => [4,2,2,1,1] => [3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => 3
[1,0,1,1,0,1,1,0,1,0,0,0] => [3,2,2,1,1] => [3,3,3] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,2,2,1,1] => [5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => 4
[1,0,1,1,1,0,1,0,1,0,0,0] => [3,2,1,1,1] => [3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => 5
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,2,1,1,1] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 3
[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,1,1,1] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,0,1,1,1,1,0,1,0,0,0,0] => [2,1,1,1,1] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 3
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => [3,2] => [1,0,1,1,1,0,0,0] => 2
[1,1,0,0,1,1,1,0,0,1,0,0] => [4,2,2,2] => [5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => 4
[1,1,0,1,0,1,0,1,1,0,0,0] => [3,3,2,1] => [3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => 3
[1,1,0,1,0,1,1,0,1,0,0,0] => [3,2,2,1] => [3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => 5
[1,1,0,1,1,0,1,0,0,1,0,0] => [4,2,1,1] => [4,4] => [1,1,1,0,1,0,1,0,0,0] => 3
[1,1,0,1,1,0,1,0,1,0,0,0] => [3,2,1,1] => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => 4
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,2,1,1] => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 5
[1,1,0,1,1,1,0,0,0,1,0,0] => [4,1,1,1] => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 5
[1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,1,1] => [3,3] => [1,1,1,0,1,0,0,0] => 2
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,1,1,1] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 4
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1] => [3,1] => [1,0,1,0,1,1,0,0] => 3
[1,1,1,0,0,0,1,1,0,1,0,0] => [4,3,3] => [2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => 3
[1,1,1,0,0,0,1,1,1,0,0,0] => [3,3,3] => [2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => 5
[1,1,1,0,0,1,0,1,1,0,0,0] => [3,3,2] => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => 2
[1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,2] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => 4
[1,1,1,0,0,1,1,1,0,0,0,0] => [2,2,2] => [2,2,2] => [1,1,1,1,0,0,0,0] => 1
[1,1,1,0,1,0,1,0,1,0,0,0] => [3,2,1] => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5
[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => 3
[1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,1] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 5
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,1,1] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 4
[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => [2,2] => [1,1,1,0,0,0] => 1
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [2,1] => [1,0,1,1,0,0] => 2
[1,1,1,1,0,0,0,1,0,0,1,0] => [5,3] => [2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => 5
[1,1,1,1,0,0,0,1,1,0,0,0] => [3,3] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
[1,1,1,1,0,0,1,0,0,0,1,0] => [5,2] => [2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => 5
[1,1,1,1,0,0,1,0,0,1,0,0] => [4,2] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 4
[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => [5] => [1,0,1,0,1,0,1,0,1,0] => 5
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [4] => [1,0,1,0,1,0,1,0] => 4
[1,1,1,1,0,1,0,0,0,0,1,0] => [5,1] => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 5
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 4
[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 3
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [3] => [1,0,1,0,1,0] => 3
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [2] => [1,0,1,0] => 2
[1,1,1,1,1,0,0,0,0,0,1,0] => [5] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 4
[1,1,1,1,1,0,0,0,0,1,0,0] => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 3
[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => [1,1,1] => [1,1,0,1,0,0] => 2
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [1,1] => [1,1,0,0] => 1
[1,1,1,1,1,0,1,0,0,0,0,0] => [1] => [1] => [1,0] => 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [2,2,2,2,2,1] => [3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
[1,0,1,1,0,1,1,1,1,0,0,0,0,0] => [2,2,2,2,1,1] => [3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => 5
[1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [2,2,2,1,1,1] => [3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => 5
[1,0,1,1,1,1,0,0,1,0,0,1,0,0] => [5,3,1,1,1,1] => [4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
[1,0,1,1,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1,1,1] => [3,3,2] => [1,1,1,0,1,1,0,0,0,0] => 2
[1,0,1,1,1,1,1,0,0,0,1,0,0,0] => [4,1,1,1,1,1] => [4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => 5
[1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1,1,1] => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 4
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Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in DyckPaths/NakayamaAlgebras. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in DyckPaths/NakayamaAlgebras. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
Loehr-Warrington
Description
Return a partition whose diagonal inversion number is the length of the preimage.
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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