- FindStat

Identifier

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St001657: Integer partitions ⟶ ℤ

Values

[1,0,1,0] => [1] => [1] => 0

[1,0,1,0,1,0] => [2,1] => [1,1,1] => 0

[1,0,1,1,0,0] => [1,1] => [2] => 1

[1,1,0,0,1,0] => [2] => [1,1] => 0

[1,1,0,1,0,0] => [1] => [1] => 0

[1,0,1,0,1,0,1,0] => [3,2,1] => [3,1,1,1] => 0

[1,0,1,0,1,1,0,0] => [2,2,1] => [4,1] => 0

[1,0,1,1,0,0,1,0] => [3,1,1] => [3,2] => 1

[1,0,1,1,0,1,0,0] => [2,1,1] => [2,1,1] => 1

[1,0,1,1,1,0,0,0] => [1,1,1] => [2,1] => 1

[1,1,0,0,1,0,1,0] => [3,2] => [3,1,1] => 0

[1,1,0,0,1,1,0,0] => [2,2] => [4] => 0

[1,1,0,1,0,0,1,0] => [3,1] => [3,1] => 0

[1,1,0,1,0,1,0,0] => [2,1] => [1,1,1] => 0

[1,1,0,1,1,0,0,0] => [1,1] => [2] => 1

[1,1,1,0,0,0,1,0] => [3] => [3] => 0

[1,1,1,0,0,1,0,0] => [2] => [1,1] => 0

[1,1,1,0,1,0,0,0] => [1] => [1] => 0

[1,0,1,0,1,0,1,0,1,0] => [4,3,2,1] => [3,1,1,1,1,1,1,1] => 0

[1,0,1,0,1,0,1,1,0,0] => [3,3,2,1] => [6,1,1,1] => 0

[1,0,1,0,1,1,0,0,1,0] => [4,2,2,1] => [4,1,1,1,1,1] => 0

[1,0,1,0,1,1,0,1,0,0] => [3,2,2,1] => [4,3,1] => 0

[1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => [4,1,1,1] => 0

[1,0,1,1,0,0,1,0,1,0] => [4,3,1,1] => [3,2,1,1,1,1] => 1

[1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => [6,2] => 1

[1,0,1,1,0,1,0,0,1,0] => [4,2,1,1] => [2,1,1,1,1,1,1] => 1

[1,0,1,1,0,1,0,1,0,0] => [3,2,1,1] => [3,2,1,1] => 1

[1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => [4,2] => 1

[1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => [2,1,1,1,1,1] => 1

[1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => [3,2,1] => 1

[1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => [2,1,1,1] => 1

[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => [2,2] => 2

[1,1,0,0,1,0,1,0,1,0] => [4,3,2] => [3,1,1,1,1,1,1] => 0

[1,1,0,0,1,0,1,1,0,0] => [3,3,2] => [6,1,1] => 0

[1,1,0,0,1,1,0,0,1,0] => [4,2,2] => [4,1,1,1,1] => 0

[1,1,0,0,1,1,0,1,0,0] => [3,2,2] => [4,3] => 0

[1,1,0,0,1,1,1,0,0,0] => [2,2,2] => [4,1,1] => 0

[1,1,0,1,0,0,1,0,1,0] => [4,3,1] => [3,1,1,1,1,1] => 0

[1,1,0,1,0,0,1,1,0,0] => [3,3,1] => [6,1] => 0

[1,1,0,1,0,1,0,0,1,0] => [4,2,1] => [1,1,1,1,1,1,1] => 0

[1,1,0,1,0,1,0,1,0,0] => [3,2,1] => [3,1,1,1] => 0

[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [4,1] => 0

[1,1,0,1,1,0,0,0,1,0] => [4,1,1] => [2,1,1,1,1] => 1

[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => [3,2] => 1

[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => [2,1,1] => 1

[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [2,1] => 1

[1,1,1,0,0,0,1,0,1,0] => [4,3] => [3,1,1,1,1] => 0

[1,1,1,0,0,0,1,1,0,0] => [3,3] => [6] => 0

[1,1,1,0,0,1,0,0,1,0] => [4,2] => [1,1,1,1,1,1] => 0

[1,1,1,0,0,1,0,1,0,0] => [3,2] => [3,1,1] => 0

[1,1,1,0,0,1,1,0,0,0] => [2,2] => [4] => 0

[1,1,1,0,1,0,0,0,1,0] => [4,1] => [1,1,1,1,1] => 0

[1,1,1,0,1,0,0,1,0,0] => [3,1] => [3,1] => 0

[1,1,1,0,1,0,1,0,0,0] => [2,1] => [1,1,1] => 0

[1,1,1,0,1,1,0,0,0,0] => [1,1] => [2] => 1

[1,1,1,1,0,0,0,0,1,0] => [4] => [1,1,1,1] => 0

[1,1,1,1,0,0,0,1,0,0] => [3] => [3] => 0

[1,1,1,1,0,0,1,0,0,0] => [2] => [1,1] => 0

[1,1,1,1,0,1,0,0,0,0] => [1] => [1] => 0

[1,0,1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [5,3,1,1,1,1,1,1,1] => 0

[1,0,1,0,1,0,1,0,1,1,0,0] => [4,4,3,2,1] => [8,3,1,1,1] => 0

[1,0,1,0,1,0,1,1,0,0,1,0] => [5,3,3,2,1] => [6,5,1,1,1] => 0

[1,0,1,0,1,0,1,1,0,1,0,0] => [4,3,3,2,1] => [6,1,1,1,1,1,1,1] => 0

[1,0,1,0,1,0,1,1,1,0,0,0] => [3,3,3,2,1] => [6,3,1,1,1] => 0

[1,0,1,0,1,1,0,0,1,0,1,0] => [5,4,2,2,1] => [5,4,1,1,1,1,1] => 0

[1,0,1,0,1,1,0,0,1,1,0,0] => [4,4,2,2,1] => [8,4,1] => 0

[1,0,1,0,1,1,0,1,0,0,1,0] => [5,3,2,2,1] => [5,4,3,1] => 0

[1,0,1,0,1,1,0,1,0,1,0,0] => [4,3,2,2,1] => [4,3,1,1,1,1,1] => 0

[1,0,1,0,1,1,0,1,1,0,0,0] => [3,3,2,2,1] => [6,4,1] => 0

[1,0,1,0,1,1,1,0,0,0,1,0] => [5,2,2,2,1] => [5,4,1,1,1] => 0

[1,0,1,0,1,1,1,0,0,1,0,0] => [4,2,2,2,1] => [4,1,1,1,1,1,1,1] => 0

[1,0,1,0,1,1,1,0,1,0,0,0] => [3,2,2,2,1] => [4,3,1,1,1] => 0

[1,0,1,0,1,1,1,1,0,0,0,0] => [2,2,2,2,1] => [4,4,1] => 0

[1,0,1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,1] => [5,3,2,1,1,1,1] => 1

[1,0,1,1,0,0,1,0,1,1,0,0] => [4,4,3,1,1] => [8,3,2] => 1

[1,0,1,1,0,0,1,1,0,0,1,0] => [5,3,3,1,1] => [6,5,2] => 1

[1,0,1,1,0,0,1,1,0,1,0,0] => [4,3,3,1,1] => [6,2,1,1,1,1] => 1

[1,0,1,1,0,0,1,1,1,0,0,0] => [3,3,3,1,1] => [6,3,2] => 1

[1,0,1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,1] => [5,2,1,1,1,1,1,1] => 1

[1,0,1,1,0,1,0,0,1,1,0,0] => [4,4,2,1,1] => [8,2,1,1] => 1

[1,0,1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,1] => [5,3,2,1,1] => 1

[1,0,1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,1] => [3,2,1,1,1,1,1,1] => 1

[1,0,1,1,0,1,0,1,1,0,0,0] => [3,3,2,1,1] => [6,2,1,1] => 1

[1,0,1,1,0,1,1,0,0,0,1,0] => [5,2,2,1,1] => [5,4,2] => 1

[1,0,1,1,0,1,1,0,0,1,0,0] => [4,2,2,1,1] => [4,2,1,1,1,1] => 1

[1,0,1,1,0,1,1,0,1,0,0,0] => [3,2,2,1,1] => [4,3,2] => 1

[1,0,1,1,0,1,1,1,0,0,0,0] => [2,2,2,1,1] => [4,2,1,1] => 1

[1,0,1,1,1,0,0,0,1,0,1,0] => [5,4,1,1,1] => [5,2,1,1,1,1,1] => 1

[1,0,1,1,1,0,0,0,1,1,0,0] => [4,4,1,1,1] => [8,2,1] => 1

[1,0,1,1,1,0,0,1,0,0,1,0] => [5,3,1,1,1] => [5,3,2,1] => 1

[1,0,1,1,1,0,0,1,0,1,0,0] => [4,3,1,1,1] => [3,2,1,1,1,1,1] => 1

[1,0,1,1,1,0,0,1,1,0,0,0] => [3,3,1,1,1] => [6,2,1] => 1

[1,0,1,1,1,0,1,0,0,0,1,0] => [5,2,1,1,1] => [5,2,1,1,1] => 1

[1,0,1,1,1,0,1,0,0,1,0,0] => [4,2,1,1,1] => [2,1,1,1,1,1,1,1] => 1

[1,0,1,1,1,0,1,0,1,0,0,0] => [3,2,1,1,1] => [3,2,1,1,1] => 1

[1,0,1,1,1,0,1,1,0,0,0,0] => [2,2,1,1,1] => [4,2,1] => 1

[1,0,1,1,1,1,0,0,0,0,1,0] => [5,1,1,1,1] => [5,2,2] => 2

[1,0,1,1,1,1,0,0,0,1,0,0] => [4,1,1,1,1] => [2,2,1,1,1,1] => 2

[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,1,1,1] => [3,2,2] => 2

[1,0,1,1,1,1,0,1,0,0,0,0] => [2,1,1,1,1] => [2,2,1,1] => 2

[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => [2,2,1] => 2

>>> Load all 1797 entries. <<<

search for individual values

searching the database for the individual values of this statistic

Description

The number of twos in an integer partition.
The total number of twos in all partitions of $n$ is equal to the total number of singletons St001484The number of singletons of an integer partition. in all partitions of $n-1$, see [1].

Map

Glaisher-Franklin inverse

Description

The Glaisher-Franklin bijection on integer partitions.
This map sends the number of distinct repeated part sizes to the number of distinct even part sizes, see [1, 3.3.1].

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 $(n-1,\ldots,2,1)$.