Identifier
-
Mp00327:
Dyck paths
—inverse Kreweras complement⟶
Dyck paths
St001291: Dyck paths ⟶ ℤ
Values
[1,0] => [1,0] => 1
[1,0,1,0] => [1,1,0,0] => 1
[1,1,0,0] => [1,0,1,0] => 2
[1,0,1,0,1,0] => [1,1,1,0,0,0] => 1
[1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
[1,1,0,0,1,0] => [1,0,1,1,0,0] => 2
[1,1,0,1,0,0] => [1,1,0,0,1,0] => 3
[1,1,1,0,0,0] => [1,0,1,0,1,0] => 3
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 1
[1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 2
[1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 2
[1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => 3
[1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 3
[1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => 2
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,0,0] => 3
[1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 3
[1,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0] => 4
[1,1,0,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => 4
[1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => 3
[1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => 4
[1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => 4
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 4
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 2
[1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 2
[1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 3
[1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => 3
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => 2
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 3
[1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => 3
[1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 4
[1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => 4
[1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 3
[1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,0,0,1,0,0] => 4
[1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,0,0,1,0,0] => 4
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 4
[1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => 3
[1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => 3
[1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 4
[1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => 4
[1,1,0,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 3
[1,1,0,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => 4
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 4
[1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 5
[1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => 5
[1,1,0,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 4
[1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 5
[1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,1,0,0,1,0] => 5
[1,1,0,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => 5
[1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0] => 3
[1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => 4
[1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => 4
[1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 5
[1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => 5
[1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => 4
[1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => 5
[1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => 5
[1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => 5
[1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => 4
[1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => 5
[1,1,1,1,0,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 5
[1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => 5
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => 3
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => 3
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => 3
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 4
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => 4
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 3
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => 4
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => 4
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => 4
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 2
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => 3
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => 3
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => 4
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => 4
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 3
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => 4
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 4
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 5
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => 5
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => 4
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => 5
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => 5
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => 5
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => 3
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => 4
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => 4
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => 5
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => 5
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => 4
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => 5
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 5
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => 5
>>> Load all 196 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
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
inverse Kreweras complement
Description
Return the inverse of the Kreweras complement of a Dyck path, regarded as a noncrossing set partition.
To identify Dyck paths and noncrossing set partitions, this maps uses the following classical bijection. The number of down steps after the $i$-th up step of the Dyck path is the size of the block of the set partition whose maximal element is $i$. If $i$ is not a maximal element of a block, the $(i+1)$-st step is also an up step.
To identify Dyck paths and noncrossing set partitions, this maps uses the following classical bijection. The number of down steps after the $i$-th up step of the Dyck path is the size of the block of the set partition whose maximal element is $i$. If $i$ is not a maximal element of a block, the $(i+1)$-st step is also an up step.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!