Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001514: Dyck paths ⟶ ℤ
Values
[1,0] => [] => [] => [1,0] => 1
[1,0,1,0] => [1] => [1,0,1,0] => [1,1,0,1,0,0] => 1
[1,1,0,0] => [] => [] => [1,0] => 1
[1,0,1,0,1,0] => [2,1] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 2
[1,0,1,1,0,0] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 2
[1,1,0,0,1,0] => [2] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 2
[1,1,0,1,0,0] => [1] => [1,0,1,0] => [1,1,0,1,0,0] => 1
[1,1,1,0,0,0] => [] => [] => [1,0] => 1
[1,0,1,0,1,0,1,0] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3
[1,0,1,0,1,1,0,0] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 3
[1,0,1,1,0,0,1,0] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 3
[1,0,1,1,0,1,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 2
[1,0,1,1,1,0,0,0] => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 2
[1,1,0,0,1,0,1,0] => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 3
[1,1,0,0,1,1,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => 3
[1,1,0,1,0,0,1,0] => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => 2
[1,1,0,1,0,1,0,0] => [2,1] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 2
[1,1,0,1,1,0,0,0] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 2
[1,1,1,0,0,0,1,0] => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[1,1,1,0,0,1,0,0] => [2] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 2
[1,1,1,0,1,0,0,0] => [1] => [1,0,1,0] => [1,1,0,1,0,0] => 1
[1,1,1,1,0,0,0,0] => [] => [] => [1,0] => 1
[1,1,0,1,0,1,0,1,0,0] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3
[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 3
[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 3
[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 2
[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 2
[1,1,1,0,0,1,0,1,0,0] => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 3
[1,1,1,0,0,1,1,0,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => 3
[1,1,1,0,1,0,0,1,0,0] => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => 2
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 2
[1,1,1,0,1,1,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 2
[1,1,1,1,0,0,0,1,0,0] => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[1,1,1,1,0,0,1,0,0,0] => [2] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 2
[1,1,1,1,0,1,0,0,0,0] => [1] => [1,0,1,0] => [1,1,0,1,0,0] => 1
[1,1,1,1,1,0,0,0,0,0] => [] => [] => [1,0] => 1
[1,1,1,0,1,0,1,0,1,0,0,0] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3
[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 3
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 3
[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 2
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 2
[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 3
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => 3
[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => 2
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 2
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 2
[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 2
[1,1,1,1,1,0,1,0,0,0,0,0] => [1] => [1,0,1,0] => [1,1,0,1,0,0] => 1
[1,1,1,1,1,1,0,0,0,0,0,0] => [] => [] => [1,0] => 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3
[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 3
[1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 3
[1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 2
[1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 3
[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => 3
[1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 2
[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 2
[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [2] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [1] => [1,0,1,0] => [1,1,0,1,0,0] => 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [] => [] => [1,0] => 1
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 3
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 3
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 2
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0] => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 2
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0] => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 3
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => 3
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0] => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => 2
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 2
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 2
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0] => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [2] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 2
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [1] => [1,0,1,0] => [1,1,0,1,0,0] => 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [] => [] => [1,0] => 1
[] => [] => [] => [1,0] => 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 3
[1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 3
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 2
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 2
[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0] => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 3
[1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => 3
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0] => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => 2
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 2
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 2
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0] => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0] => [2] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 2
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [1] => [1,0,1,0] => [1,1,0,1,0,0] => 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [] => [] => [1,0] => 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [] => [] => [1,0] => 1
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 2
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0] => [2] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 2
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 2
[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 3
[1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0] => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 3
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0] => [2] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 2
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0] => [1] => [1,0,1,0] => [1,1,0,1,0,0] => 1
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Description
The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule.
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down 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.
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