Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000907: Posets ⟶ ℤ
Values
[1,0,1,0] => [1] => [[1],[]] => ([],1) => 1
[1,0,1,0,1,0] => [2,1] => [[2,1],[]] => ([(0,1),(0,2)],3) => 1
[1,0,1,1,0,0] => [1,1] => [[1,1],[]] => ([(0,1)],2) => 2
[1,1,0,0,1,0] => [2] => [[2],[]] => ([(0,1)],2) => 2
[1,1,0,1,0,0] => [1] => [[1],[]] => ([],1) => 1
[1,0,1,0,1,0,1,0] => [3,2,1] => [[3,2,1],[]] => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6) => 1
[1,0,1,0,1,1,0,0] => [2,2,1] => [[2,2,1],[]] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 1
[1,0,1,1,0,0,1,0] => [3,1,1] => [[3,1,1],[]] => ([(0,3),(0,4),(3,2),(4,1)],5) => 1
[1,0,1,1,0,1,0,0] => [2,1,1] => [[2,1,1],[]] => ([(0,2),(0,3),(3,1)],4) => 1
[1,0,1,1,1,0,0,0] => [1,1,1] => [[1,1,1],[]] => ([(0,2),(2,1)],3) => 3
[1,1,0,0,1,0,1,0] => [3,2] => [[3,2],[]] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 1
[1,1,0,0,1,1,0,0] => [2,2] => [[2,2],[]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,1,0,1,0,0,1,0] => [3,1] => [[3,1],[]] => ([(0,2),(0,3),(3,1)],4) => 1
[1,1,0,1,0,1,0,0] => [2,1] => [[2,1],[]] => ([(0,1),(0,2)],3) => 1
[1,1,0,1,1,0,0,0] => [1,1] => [[1,1],[]] => ([(0,1)],2) => 2
[1,1,1,0,0,0,1,0] => [3] => [[3],[]] => ([(0,2),(2,1)],3) => 3
[1,1,1,0,0,1,0,0] => [2] => [[2],[]] => ([(0,1)],2) => 2
[1,1,1,0,1,0,0,0] => [1] => [[1],[]] => ([],1) => 1
[1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => [[2,2,1,1],[]] => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6) => 1
[1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => [[4,1,1,1],[]] => ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7) => 1
[1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => [[3,1,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6) => 1
[1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => [[2,1,1,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => 1
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => [[1,1,1,1],[]] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,0,0,1,1,1,0,0,0] => [2,2,2] => [[2,2,2],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[1,1,0,1,0,1,0,1,0,0] => [3,2,1] => [[3,2,1],[]] => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6) => 1
[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [[2,2,1],[]] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 1
[1,1,0,1,1,0,0,0,1,0] => [4,1,1] => [[4,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6) => 1
[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => [[3,1,1],[]] => ([(0,3),(0,4),(3,2),(4,1)],5) => 1
[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => [[2,1,1],[]] => ([(0,2),(0,3),(3,1)],4) => 1
[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [[1,1,1],[]] => ([(0,2),(2,1)],3) => 3
[1,1,1,0,0,0,1,1,0,0] => [3,3] => [[3,3],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[1,1,1,0,0,1,0,0,1,0] => [4,2] => [[4,2],[]] => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6) => 1
[1,1,1,0,0,1,0,1,0,0] => [3,2] => [[3,2],[]] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 1
[1,1,1,0,0,1,1,0,0,0] => [2,2] => [[2,2],[]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,1,1,0,1,0,0,0,1,0] => [4,1] => [[4,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => 1
[1,1,1,0,1,0,0,1,0,0] => [3,1] => [[3,1],[]] => ([(0,2),(0,3),(3,1)],4) => 1
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [[2,1],[]] => ([(0,1),(0,2)],3) => 1
[1,1,1,0,1,1,0,0,0,0] => [1,1] => [[1,1],[]] => ([(0,1)],2) => 2
[1,1,1,1,0,0,0,0,1,0] => [4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,1,1,0,0,0,1,0,0] => [3] => [[3],[]] => ([(0,2),(2,1)],3) => 3
[1,1,1,1,0,0,1,0,0,0] => [2] => [[2],[]] => ([(0,1)],2) => 2
[1,1,1,1,0,1,0,0,0,0] => [1] => [[1],[]] => ([],1) => 1
[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,1,1,1] => [[3,1,1,1,1],[]] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7) => 1
[1,0,1,1,1,1,0,1,0,0,0,0] => [2,1,1,1,1] => [[2,1,1,1,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => 1
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => [[1,1,1,1,1],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,2,1,1] => [[2,2,1,1],[]] => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6) => 1
[1,1,0,1,1,1,0,0,0,1,0,0] => [4,1,1,1] => [[4,1,1,1],[]] => ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7) => 1
[1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,1,1] => [[3,1,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6) => 1
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,1,1,1] => [[2,1,1,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => 1
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1] => [[1,1,1,1],[]] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,1,0,0,1,1,1,0,0,0,0] => [2,2,2] => [[2,2,2],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[1,1,1,0,1,0,1,0,1,0,0,0] => [3,2,1] => [[3,2,1],[]] => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6) => 1
[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => [[2,2,1],[]] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 1
[1,1,1,0,1,1,0,0,0,0,1,0] => [5,1,1] => [[5,1,1],[]] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7) => 1
[1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,1] => [[4,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6) => 1
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,1,1] => [[3,1,1],[]] => ([(0,3),(0,4),(3,2),(4,1)],5) => 1
[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => [[2,1,1],[]] => ([(0,2),(0,3),(3,1)],4) => 1
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [[1,1,1],[]] => ([(0,2),(2,1)],3) => 3
[1,1,1,1,0,0,0,1,1,0,0,0] => [3,3] => [[3,3],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[1,1,1,1,0,0,1,0,0,1,0,0] => [4,2] => [[4,2],[]] => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6) => 1
[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => [[3,2],[]] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 1
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [[2,2],[]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,1,1,1,0,1,0,0,0,0,1,0] => [5,1] => [[5,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => 1
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,1] => [[4,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => 1
[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [[3,1],[]] => ([(0,2),(0,3),(3,1)],4) => 1
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [[2,1],[]] => ([(0,1),(0,2)],3) => 1
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [[1,1],[]] => ([(0,1)],2) => 2
[1,1,1,1,1,0,0,0,0,0,1,0] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,1,1,0,0,0,0,1,0,0] => [4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => [[3],[]] => ([(0,2),(2,1)],3) => 3
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [[2],[]] => ([(0,1)],2) => 2
[1,1,1,1,1,0,1,0,0,0,0,0] => [1] => [[1],[]] => ([],1) => 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1,1,1] => [[2,1,1,1,1,1],[]] => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7) => 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,0,1,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1,1] => [[3,1,1,1,1],[]] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7) => 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1,1] => [[2,1,1,1,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1] => [[1,1,1,1,1],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,0,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1] => [[2,2,1,1],[]] => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6) => 1
[1,1,1,0,1,1,1,0,0,0,1,0,0,0] => [4,1,1,1] => [[4,1,1,1],[]] => ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7) => 1
[1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1] => [[3,1,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6) => 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1] => [[2,1,1,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1] => [[1,1,1,1],[]] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [2,2,2] => [[2,2,2],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [3,2,1] => [[3,2,1],[]] => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6) => 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [2,2,1] => [[2,2,1],[]] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 1
[1,1,1,1,0,1,1,0,0,0,0,1,0,0] => [5,1,1] => [[5,1,1],[]] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7) => 1
[1,1,1,1,0,1,1,0,0,0,1,0,0,0] => [4,1,1] => [[4,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6) => 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [3,1,1] => [[3,1,1],[]] => ([(0,3),(0,4),(3,2),(4,1)],5) => 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [2,1,1] => [[2,1,1],[]] => ([(0,2),(0,3),(3,1)],4) => 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [1,1,1] => [[1,1,1],[]] => ([(0,2),(2,1)],3) => 3
[1,1,1,1,1,0,0,0,1,1,0,0,0,0] => [3,3] => [[3,3],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[1,1,1,1,1,0,0,1,0,0,1,0,0,0] => [4,2] => [[4,2],[]] => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6) => 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [3,2] => [[3,2],[]] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [2,2] => [[2,2],[]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [6,1] => [[6,1],[]] => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7) => 1
[1,1,1,1,1,0,1,0,0,0,0,1,0,0] => [5,1] => [[5,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => 1
[1,1,1,1,1,0,1,0,0,0,1,0,0,0] => [4,1] => [[4,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => 1
[1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [3,1] => [[3,1],[]] => ([(0,2),(0,3),(3,1)],4) => 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [2,1] => [[2,1],[]] => ([(0,1),(0,2)],3) => 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,1] => [[1,1],[]] => ([(0,1)],2) => 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
>>> Load all 222 entries. <<<
search for individual values
searching the database for the individual values of this statistic
Description
The number of maximal antichains of minimal length in a poset.
Map
to skew partition
Description
The partition regarded as a skew partition.
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
cell poset
Description
The Young diagram of a skew partition regarded as a poset.
This is the poset on the cells of the Young diagram, such that a cell $d$ is greater than a cell $c$ if the entry in $d$ must be larger than the entry of $c$ in any standard Young tableau on the skew partition.
This is the poset on the cells of the Young diagram, such that a cell $d$ is greater than a cell $c$ if the entry in $d$ must be larger than the entry of $c$ in any standard Young tableau on the skew partition.
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!