- FindStat

Identifier

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1734 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 2\ q + 2\ q^{2}$

$F_{4} = 1 + 3\ q + 5\ q^{2} + 5\ q^{3}$

$F_{5} = 1 + 4\ q + 9\ q^{2} + 14\ q^{3} + 14\ q^{4}$

$F_{6} = 1 + 5\ q + 14\ q^{2} + 28\ q^{3} + 42\ q^{4} + 42\ q^{5}$

Description

The largest part of an integer 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

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

Map

conjugate

Description

Return the conjugate partition of the partition.
The conjugate partition of the partition

$\lambda$ of

$n$ is the partition

$\lambda^*$ whose Ferrers diagram is obtained from the diagram of

$\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.