Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤ
Values
[1,0,1,0,1,0] => [2,1] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 1
[1,0,1,1,0,0] => [1,1] => [1,1,0,0] => [1,1,1,0,0,0] => 1
[1,1,0,0,1,0] => [2] => [1,0,1,0] => [1,1,0,1,0,0] => 1
[1,0,1,0,1,1,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 1
[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] => 1
[1,0,1,1,1,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 1
[1,1,0,0,1,0,1,0] => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 1
[1,1,0,0,1,1,0,0] => [2,2] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => 1
[1,1,0,1,0,0,1,0] => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 1
[1,1,0,1,0,1,0,0] => [2,1] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 1
[1,1,0,1,1,0,0,0] => [1,1] => [1,1,0,0] => [1,1,1,0,0,0] => 1
[1,1,1,0,0,0,1,0] => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 1
[1,1,1,0,0,1,0,0] => [2] => [1,0,1,0] => [1,1,0,1,0,0] => 1
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => 1
[1,1,0,0,1,1,1,0,0,0] => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 1
[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] => 1
[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 1
[1,1,1,0,0,0,1,1,0,0] => [3,3] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 1
[1,1,1,0,0,1,0,1,0,0] => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 1
[1,1,1,0,0,1,1,0,0,0] => [2,2] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => 1
[1,1,1,0,1,0,0,1,0,0] => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 1
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 1
[1,1,1,0,1,1,0,0,0,0] => [1,1] => [1,1,0,0] => [1,1,1,0,0,0] => 1
[1,1,1,1,0,0,0,0,1,0] => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1
[1,1,1,1,0,0,0,1,0,0] => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 1
[1,1,1,1,0,0,1,0,0,0] => [2] => [1,0,1,0] => [1,1,0,1,0,0] => 1
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => 1
[1,1,1,0,0,1,1,1,0,0,0,0] => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 1
[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] => 1
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 1
[1,1,1,1,0,0,0,1,1,0,0,0] => [3,3] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 1
[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 1
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => 1
[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 1
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 1
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [1,1,0,0] => [1,1,1,0,0,0] => 1
[1,1,1,1,1,0,0,0,0,1,0,0] => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1
[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 1
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [1,0,1,0] => [1,1,0,1,0,0] => 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 1
[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] => 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 1
[1,1,1,1,1,0,0,0,1,1,0,0,0,0] => [3,3] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [2,2] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => 1
[1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [2,1] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,1] => [1,1,0,0] => [1,1,1,0,0,0] => 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [2] => [1,0,1,0] => [1,1,0,1,0,0] => 1
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => 1
[1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0] => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 1
[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] => 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 1
[1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0] => [3,3] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 1
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0] => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 1
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0] => [2,2] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => 1
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0] => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [2,1] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [1,1] => [1,1,0,0] => [1,1,1,0,0,0] => 1
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0] => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [2] => [1,0,1,0] => [1,1,0,1,0,0] => 1
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => 1
[1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0] => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 1
[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] => 1
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 1
[1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0] => [3,3] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 1
[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0] => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 1
[1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0] => [2,2] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => 1
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0] => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [2,1] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0] => [1,1] => [1,1,0,0] => [1,1,1,0,0,0] => 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0] => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0] => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0] => [2] => [1,0,1,0] => [1,1,0,1,0,0] => 1
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0] => [1,1] => [1,1,0,0] => [1,1,1,0,0,0] => 1
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0] => [2] => [1,0,1,0] => [1,1,0,1,0,0] => 1
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0] => [1,1] => [1,1,0,0] => [1,1,1,0,0,0] => 1
[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 1
[1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0] => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 1
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0] => [2] => [1,0,1,0] => [1,1,0,1,0,0] => 1
[1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 1
[1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0] => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1
[1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0] => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 1
[1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0] => [2,2] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => 1
[1,1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,0] => [3,3] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 1
[1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 1
[1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0] => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 1
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0] => [2,1] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 1
[1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0] => [2,1] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0] => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0,0] => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 1
>>> Load all 108 entries. <<<
search for individual values
searching the database for the individual values of this statistic
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
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
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
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down 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!