Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001526: Dyck paths ⟶ ℤ
Values
[1,0,1,0] => [1] => [1,0,1,0] => [1,1,0,0] => 2
[1,0,1,0,1,0] => [2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 3
[1,0,1,1,0,0] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 3
[1,1,0,0,1,0] => [2] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => 2
[1,1,0,1,0,0] => [1] => [1,0,1,0] => [1,1,0,0] => 2
[1,0,1,0,1,0,1,0] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 4
[1,0,1,0,1,1,0,0] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 4
[1,0,1,1,0,0,1,0] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 3
[1,0,1,1,0,1,0,0] => [2,1,1] => [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] => [1,1,1] => [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] => [3,2] => [1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => 3
[1,1,0,0,1,1,0,0] => [2,2] => [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] => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 2
[1,1,0,1,0,1,0,0] => [2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 3
[1,1,0,1,1,0,0,0] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 3
[1,1,1,0,0,0,1,0] => [3] => [1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => 2
[1,1,1,0,0,1,0,0] => [2] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => 2
[1,1,1,0,1,0,0,0] => [1] => [1,0,1,0] => [1,1,0,0] => 2
[1,0,1,0,1,0,1,0,1,0] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 5
[1,0,1,0,1,0,1,1,0,0] => [3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 5
[1,0,1,0,1,1,0,0,1,0] => [4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 4
[1,0,1,0,1,1,0,1,0,0] => [3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 4
[1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => 4
[1,0,1,1,0,0,1,0,1,0] => [4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => 4
[1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 4
[1,0,1,1,0,1,0,0,1,0] => [4,2,1,1] => [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] => [3,2,1,1] => [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] => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => 3
[1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => [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] => [3,1,1,1] => [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] => [2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,0,0,1,0,0] => 3
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 3
[1,1,0,0,1,0,1,0,1,0] => [4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => 4
[1,1,0,0,1,0,1,1,0,0] => [3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => 4
[1,1,0,0,1,1,0,0,1,0] => [4,2,2] => [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] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 3
[1,1,0,0,1,1,1,0,0,0] => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => 3
[1,1,0,1,0,0,1,0,1,0] => [4,3,1] => [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] => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => 3
[1,1,0,1,0,1,0,0,1,0] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 3
[1,1,0,1,0,1,0,1,0,0] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 4
[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 4
[1,1,0,1,1,0,0,0,1,0] => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 2
[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 3
[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => 3
[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 3
[1,1,1,0,0,0,1,0,1,0] => [4,3] => [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] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => 3
[1,1,1,0,0,1,0,0,1,0] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => 3
[1,1,1,0,0,1,0,1,0,0] => [3,2] => [1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => 3
[1,1,1,0,0,1,1,0,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,0,0] => 3
[1,1,1,0,1,0,0,0,1,0] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => 2
[1,1,1,0,1,0,0,1,0,0] => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 2
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 3
[1,1,1,0,1,1,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 3
[1,1,1,1,0,0,0,0,1,0] => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => 2
[1,1,1,1,0,0,0,1,0,0] => [3] => [1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => 2
[1,1,1,1,0,0,1,0,0,0] => [2] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => 2
[1,1,1,1,0,1,0,0,0,0] => [1] => [1,0,1,0] => [1,1,0,0] => 2
[1,1,0,1,0,1,0,1,0,1,0,0] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 5
[1,1,0,1,0,1,0,1,1,0,0,0] => [3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 5
[1,1,0,1,0,1,1,0,0,1,0,0] => [4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 4
[1,1,0,1,0,1,1,0,1,0,0,0] => [3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 4
[1,1,0,1,0,1,1,1,0,0,0,0] => [2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => 4
[1,1,0,1,1,0,0,1,0,1,0,0] => [4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => 4
[1,1,0,1,1,0,0,1,1,0,0,0] => [3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 4
[1,1,0,1,1,0,1,0,0,1,0,0] => [4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => 3
[1,1,0,1,1,0,1,0,1,0,0,0] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 4
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => 3
[1,1,0,1,1,1,0,0,0,1,0,0] => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 3
[1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,0,0,1,0,0] => 4
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,0,0,1,0,0] => 3
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 3
[1,1,1,0,0,1,0,1,0,1,0,0] => [4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => 4
[1,1,1,0,0,1,0,1,1,0,0,0] => [3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => 4
[1,1,1,0,0,1,1,0,0,1,0,0] => [4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => 3
[1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 3
[1,1,1,0,0,1,1,1,0,0,0,0] => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => 3
[1,1,1,0,1,0,0,1,0,1,0,0] => [4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 3
[1,1,1,0,1,0,0,1,1,0,0,0] => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => 3
[1,1,1,0,1,0,1,0,0,1,0,0] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 3
[1,1,1,0,1,0,1,0,1,0,0,0] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 4
[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 4
[1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 2
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 3
[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => 3
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 3
[1,1,1,1,0,0,0,1,0,1,0,0] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0] => 3
[1,1,1,1,0,0,0,1,1,0,0,0] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => 3
[1,1,1,1,0,0,1,0,0,1,0,0] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => 3
[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => [1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => 3
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,0,0] => 3
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => 2
[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 2
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 3
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 3
[1,1,1,1,1,0,0,0,0,1,0,0] => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => 2
[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => [1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => 2
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => 2
[1,1,1,1,1,0,1,0,0,0,0,0] => [1] => [1,0,1,0] => [1,1,0,0] => 2
[1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 5
>>> Load all 264 entries. <<<
search for individual values
searching the database for the individual values of this statistic
Description
The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.
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.
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 Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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!