- FindStat

Identifier

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000143: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 1

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

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

[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] => 0

[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] => 0

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 2

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

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

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

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

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

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

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

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

[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] => 1

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

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

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

[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] => 0

[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] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 8 + 4\ q + 2\ q^{2}$

$F_{5} = 16 + 12\ q + 9\ q^{2} + 5\ q^{3}$

$F_{6} = 32 + 32\ q + 30\ q^{2} + 24\ q^{3} + 14\ q^{4}$

Description

The largest repeated part of a partition.
If the parts of the partition are all distinct, the value of the statistic is defined to be zero.

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.

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

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