- FindStat

Identifier

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000063: Integer partitions ⟶ ℤ

Values

[] => [] => [] => 1

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

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

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

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

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

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

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

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

[[],[],[],[]] => [1,0,1,0,1,0,1,0] => [3,2,1] => 24

[[],[],[[]]] => [1,0,1,0,1,1,0,0] => [2,2,1] => 12

[[],[[]],[]] => [1,0,1,1,0,0,1,0] => [3,1,1] => 12

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

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

[[[]],[],[]] => [1,1,0,0,1,0,1,0] => [3,2] => 12

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

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

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

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

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

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

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

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

[[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0] => [4,3,2,1] => 120

[[],[],[],[[]]] => [1,0,1,0,1,0,1,1,0,0] => [3,3,2,1] => 60

[[],[],[[]],[]] => [1,0,1,0,1,1,0,0,1,0] => [4,2,2,1] => 60

[[],[],[[],[]]] => [1,0,1,0,1,1,0,1,0,0] => [3,2,2,1] => 40

[[],[],[[[]]]] => [1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => 20

[[],[[]],[],[]] => [1,0,1,1,0,0,1,0,1,0] => [4,3,1,1] => 60

[[],[[]],[[]]] => [1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => 30

[[],[[],[]],[]] => [1,0,1,1,0,1,0,0,1,0] => [4,2,1,1] => 40

[[],[[[]]],[]] => [1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => 20

[[],[[],[],[]]] => [1,0,1,1,0,1,0,1,0,0] => [3,2,1,1] => 30

[[],[[],[[]]]] => [1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => 15

[[],[[[]],[]]] => [1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => 15

[[],[[[],[]]]] => [1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => 10

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

[[[]],[],[],[]] => [1,1,0,0,1,0,1,0,1,0] => [4,3,2] => 60

[[[]],[],[[]]] => [1,1,0,0,1,0,1,1,0,0] => [3,3,2] => 30

[[[]],[[]],[]] => [1,1,0,0,1,1,0,0,1,0] => [4,2,2] => 30

[[[]],[[],[]]] => [1,1,0,0,1,1,0,1,0,0] => [3,2,2] => 20

[[[]],[[[]]]] => [1,1,0,0,1,1,1,0,0,0] => [2,2,2] => 10

[[[],[]],[],[]] => [1,1,0,1,0,0,1,0,1,0] => [4,3,1] => 40

[[[[]]],[],[]] => [1,1,1,0,0,0,1,0,1,0] => [4,3] => 20

[[[],[]],[[]]] => [1,1,0,1,0,0,1,1,0,0] => [3,3,1] => 20

[[[[]]],[[]]] => [1,1,1,0,0,0,1,1,0,0] => [3,3] => 10

[[[],[],[]],[]] => [1,1,0,1,0,1,0,0,1,0] => [4,2,1] => 30

[[[],[[]]],[]] => [1,1,0,1,1,0,0,0,1,0] => [4,1,1] => 15

[[[[]],[]],[]] => [1,1,1,0,0,1,0,0,1,0] => [4,2] => 15

[[[[],[]]],[]] => [1,1,1,0,1,0,0,0,1,0] => [4,1] => 10

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

[[[],[],[],[]]] => [1,1,0,1,0,1,0,1,0,0] => [3,2,1] => 24

[[[],[],[[]]]] => [1,1,0,1,0,1,1,0,0,0] => [2,2,1] => 12

[[[],[[]],[]]] => [1,1,0,1,1,0,0,1,0,0] => [3,1,1] => 12

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

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

[[[[]],[],[]]] => [1,1,1,0,0,1,0,1,0,0] => [3,2] => 12

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

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

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

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

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

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

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

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

[[],[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 720

[[],[],[],[],[[]]] => [1,0,1,0,1,0,1,0,1,1,0,0] => [4,4,3,2,1] => 360

[[],[],[],[[]],[]] => [1,0,1,0,1,0,1,1,0,0,1,0] => [5,3,3,2,1] => 360

[[],[],[],[[],[]]] => [1,0,1,0,1,0,1,1,0,1,0,0] => [4,3,3,2,1] => 240

[[],[],[],[[[]]]] => [1,0,1,0,1,0,1,1,1,0,0,0] => [3,3,3,2,1] => 120

[[],[],[[]],[],[]] => [1,0,1,0,1,1,0,0,1,0,1,0] => [5,4,2,2,1] => 360

[[],[],[[]],[[]]] => [1,0,1,0,1,1,0,0,1,1,0,0] => [4,4,2,2,1] => 180

[[],[],[[],[]],[]] => [1,0,1,0,1,1,0,1,0,0,1,0] => [5,3,2,2,1] => 240

[[],[],[[[]]],[]] => [1,0,1,0,1,1,1,0,0,0,1,0] => [5,2,2,2,1] => 120

[[],[],[[],[],[]]] => [1,0,1,0,1,1,0,1,0,1,0,0] => [4,3,2,2,1] => 180

[[],[],[[],[[]]]] => [1,0,1,0,1,1,0,1,1,0,0,0] => [3,3,2,2,1] => 90

[[],[],[[[]],[]]] => [1,0,1,0,1,1,1,0,0,1,0,0] => [4,2,2,2,1] => 90

[[],[],[[[],[]]]] => [1,0,1,0,1,1,1,0,1,0,0,0] => [3,2,2,2,1] => 60

[[],[],[[[[]]]]] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,2,2,2,1] => 30

[[],[[]],[],[],[]] => [1,0,1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,1] => 360

[[],[[]],[],[[]]] => [1,0,1,1,0,0,1,0,1,1,0,0] => [4,4,3,1,1] => 180

[[],[[]],[[]],[]] => [1,0,1,1,0,0,1,1,0,0,1,0] => [5,3,3,1,1] => 180

[[],[[]],[[],[]]] => [1,0,1,1,0,0,1,1,0,1,0,0] => [4,3,3,1,1] => 120

[[],[[]],[[[]]]] => [1,0,1,1,0,0,1,1,1,0,0,0] => [3,3,3,1,1] => 60

[[],[[],[]],[],[]] => [1,0,1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,1] => 240

[[],[[[]]],[],[]] => [1,0,1,1,1,0,0,0,1,0,1,0] => [5,4,1,1,1] => 120

[[],[[],[]],[[]]] => [1,0,1,1,0,1,0,0,1,1,0,0] => [4,4,2,1,1] => 120

[[],[[[]]],[[]]] => [1,0,1,1,1,0,0,0,1,1,0,0] => [4,4,1,1,1] => 60

[[],[[],[],[]],[]] => [1,0,1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,1] => 180

[[],[[],[[]]],[]] => [1,0,1,1,0,1,1,0,0,0,1,0] => [5,2,2,1,1] => 90

[[],[[[]],[]],[]] => [1,0,1,1,1,0,0,1,0,0,1,0] => [5,3,1,1,1] => 90

[[],[[[],[]]],[]] => [1,0,1,1,1,0,1,0,0,0,1,0] => [5,2,1,1,1] => 60

[[],[[[[]]]],[]] => [1,0,1,1,1,1,0,0,0,0,1,0] => [5,1,1,1,1] => 30

[[],[[],[],[],[]]] => [1,0,1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,1] => 144

[[],[[],[],[[]]]] => [1,0,1,1,0,1,0,1,1,0,0,0] => [3,3,2,1,1] => 72

[[],[[],[[]],[]]] => [1,0,1,1,0,1,1,0,0,1,0,0] => [4,2,2,1,1] => 72

[[],[[],[[],[]]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => [3,2,2,1,1] => 48

[[],[[],[[[]]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => [2,2,2,1,1] => 24

[[],[[[]],[],[]]] => [1,0,1,1,1,0,0,1,0,1,0,0] => [4,3,1,1,1] => 72

[[],[[[]],[[]]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => [3,3,1,1,1] => 36

[[],[[[],[]],[]]] => [1,0,1,1,1,0,1,0,0,1,0,0] => [4,2,1,1,1] => 48

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Description

The number of linear extensions of a certain poset defined for an integer partition.
The poset is constructed in David Speyer's answer to Matt Fayers' question [3].
The value at the partition $\lambda$ also counts cover-inclusive Dyck tilings of $\lambda\setminus\mu$, summed over all $\mu$, as noticed by Philippe Nadeau in a comment.
This statistic arises in the homogeneous Garnir relations for the universal graded Specht modules for cyclotomic quiver Hecke algebras.

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

to Dyck path

Description

Return the Dyck path of the corresponding ordered tree induced by the recurrence of the Catalan numbers, see wikipedia:Catalan_number.
This sends the maximal height of the Dyck path to the depth of the tree.