- FindStat

Identifier

Mp00027: Dyck paths —to partition⟶ Integer partitions
St000384: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

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

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

$F_{5} = 1 + q + 3\ q^{2} + 9\ q^{3} + 28\ q^{4}$

$F_{6} = 1 + q + 3\ q^{2} + 9\ q^{3} + 28\ q^{4} + 90\ q^{5}$

Description

The maximal part of the shifted composition of an integer partition.
A partition

$\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding

$i-1$ to the

$i$ -th part.
The statistic is then

$\operatorname{max}_i\{ \lambda_i + i - 1 \}$ .
See also St000380Half of the maximal perimeter of a rectangle fitting into the diagram 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

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