- FindStat

Identifier

Mp00027: Dyck paths —to partition⟶ Integer partitions
St001899: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 1

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

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

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

[1,0,1,0,1,0,1,0,1,0] => [4,3,2,1] => 380

[1,0,1,0,1,0,1,1,0,0] => [3,3,2,1] => 68

[1,0,1,0,1,1,0,0,1,0] => [4,2,2,1] => 84

[1,0,1,0,1,1,0,1,0,0] => [3,2,2,1] => 33

[1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => 7

[1,0,1,1,0,0,1,0,1,0] => [4,3,1,1] => 104

[1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => 21

[1,0,1,1,0,1,0,0,1,0] => [4,2,1,1] => 46

[1,0,1,1,0,1,0,1,0,0] => [3,2,1,1] => 18

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

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

[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] => 2

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

[1,1,0,0,1,0,1,0,1,0] => [4,3,2] => 104

[1,1,0,0,1,0,1,1,0,0] => [3,3,2] => 19

[1,1,0,0,1,1,0,0,1,0] => [4,2,2] => 24

[1,1,0,0,1,1,0,1,0,0] => [3,2,2] => 10

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

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

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

[1,1,0,1,0,1,0,0,1,0] => [4,2,1] => 28

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

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

[1,1,0,1,1,0,0,0,1,0] => [4,1,1] => 9

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

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

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

[1,1,1,0,0,0,1,0,1,0] => [4,3] => 18

[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] => 9

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

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

[1,1,1,0,1,0,0,0,1,0] => [4,1] => 6

[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] => 1

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

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

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

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

[1,0,1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 84420

[1,0,1,0,1,0,1,0,1,1,0,0] => [4,4,3,2,1] => 12088

[1,0,1,0,1,0,1,1,0,0,1,0] => [5,3,3,2,1] => 12872

[1,0,1,0,1,0,1,1,0,1,0,0] => [4,3,3,2,1] => 3784

[1,0,1,0,1,0,1,1,1,0,0,0] => [3,3,3,2,1] => 410

[1,0,1,0,1,1,0,0,1,0,1,0] => [5,4,2,2,1] => 15144

[1,0,1,0,1,1,0,0,1,1,0,0] => [4,4,2,2,1] => 2245

[1,0,1,0,1,1,0,1,0,0,1,0] => [5,3,2,2,1] => 4779

[1,0,1,0,1,1,0,1,0,1,0,0] => [4,3,2,2,1] => 1460

[1,0,1,0,1,1,0,1,1,0,0,0] => [3,3,2,2,1] => 245

[1,0,1,0,1,1,1,0,0,0,1,0] => [5,2,2,2,1] => 650

[1,0,1,0,1,1,1,0,0,1,0,0] => [4,2,2,2,1] => 207

[1,0,1,0,1,1,1,0,1,0,0,0] => [3,2,2,2,1] => 75

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

[1,0,1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,1] => 19448

[1,0,1,1,0,0,1,0,1,1,0,0] => [4,4,3,1,1] => 2864

[1,0,1,1,0,0,1,1,0,0,1,0] => [5,3,3,1,1] => 3059

[1,0,1,1,0,0,1,1,0,1,0,0] => [4,3,3,1,1] => 929

[1,0,1,1,0,0,1,1,1,0,0,0] => [3,3,3,1,1] => 106

[1,0,1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,1] => 6970

[1,0,1,1,0,1,0,0,1,1,0,0] => [4,4,2,1,1] => 1060

[1,0,1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,1] => 2264

[1,0,1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,1] => 710

[1,0,1,1,0,1,0,1,1,0,0,0] => [3,3,2,1,1] => 124

[1,0,1,1,0,1,1,0,0,0,1,0] => [5,2,2,1,1] => 451

[1,0,1,1,0,1,1,0,0,1,0,0] => [4,2,2,1,1] => 148

[1,0,1,1,0,1,1,0,1,0,0,0] => [3,2,2,1,1] => 54

[1,0,1,1,0,1,1,1,0,0,0,0] => [2,2,2,1,1] => 9

[1,0,1,1,1,0,0,0,1,0,1,0] => [5,4,1,1,1] => 1196

[1,0,1,1,1,0,0,0,1,1,0,0] => [4,4,1,1,1] => 190

[1,0,1,1,1,0,0,1,0,0,1,0] => [5,3,1,1,1] => 402

[1,0,1,1,1,0,0,1,0,1,0,0] => [4,3,1,1,1] => 132

[1,0,1,1,1,0,0,1,1,0,0,0] => [3,3,1,1,1] => 25

[1,0,1,1,1,0,1,0,0,0,1,0] => [5,2,1,1,1] => 158

[1,0,1,1,1,0,1,0,0,1,0,0] => [4,2,1,1,1] => 54

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

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

[1,0,1,1,1,1,0,0,0,0,1,0] => [5,1,1,1,1] => 28

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

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

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

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

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Description

The total number of irreducible representations contained in the higher Lie character for 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 $(n-1,\ldots,2,1)$.