- FindStat

Identifier

Mp00027: Dyck paths —to partition⟶ Integer partitions
St001091: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 8 + 5\ q + q^{2}$

$F_{5} = 16 + 17\ q + 8\ q^{2} + q^{3}$

$F_{6} = 32 + 49\ q + 38\ q^{2} + 12\ q^{3} + q^{4}$

Description

The number of parts in an integer partition whose next smaller part has the same size.
In other words, this is the number of distinct parts subtracted from the number of all parts.

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

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