- FindStat

Identifier

Mp00027: Dyck paths —to partition⟶ Integer partitions
St001901: Integer partitions ⟶ ℤ

Values

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

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

[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] => 3

[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] => 1

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

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

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

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

[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] => 34

[1,0,1,0,1,0,1,1,0,0] => [3,3,2,1] => 7

[1,0,1,0,1,1,0,0,1,0] => [4,2,2,1] => 8

[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] => 1

[1,0,1,1,0,0,1,0,1,0] => [4,3,1,1] => 10

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

[1,0,1,1,0,1,0,0,1,0] => [4,2,1,1] => 6

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

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

[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] => 1

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

[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] => 10

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

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

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

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

[1,1,0,1,0,0,1,0,1,0] => [4,3,1] => 8

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

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

[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] => 1

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

[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] => 1

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

[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] => 2

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

[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] => 2

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

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

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

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

[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] => 2526

[1,0,1,0,1,0,1,0,1,1,0,0] => [4,4,3,2,1] => 397

[1,0,1,0,1,0,1,1,0,0,1,0] => [5,3,3,2,1] => 422

[1,0,1,0,1,0,1,1,0,1,0,0] => [4,3,3,2,1] => 161

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

[1,0,1,0,1,1,0,0,1,0,1,0] => [5,4,2,2,1] => 494

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

[1,0,1,0,1,1,0,1,0,0,1,0] => [5,3,2,2,1] => 187

[1,0,1,0,1,1,0,1,0,1,0,0] => [4,3,2,2,1] => 78

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

[1,0,1,0,1,1,1,0,0,0,1,0] => [5,2,2,2,1] => 32

[1,0,1,0,1,1,1,0,0,1,0,0] => [4,2,2,2,1] => 14

[1,0,1,0,1,1,1,0,1,0,0,0] => [3,2,2,2,1] => 6

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

[1,0,1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,1] => 631

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

[1,0,1,1,0,0,1,1,0,0,1,0] => [5,3,3,1,1] => 125

[1,0,1,1,0,0,1,1,0,1,0,0] => [4,3,3,1,1] => 55

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

[1,0,1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,1] => 273

[1,0,1,1,0,1,0,0,1,1,0,0] => [4,4,2,1,1] => 62

[1,0,1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,1] => 132

[1,0,1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,1] => 50

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

[1,0,1,1,0,1,1,0,0,0,1,0] => [5,2,2,1,1] => 30

[1,0,1,1,0,1,1,0,0,1,0,0] => [4,2,2,1,1] => 12

[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] => 1

[1,0,1,1,1,0,0,0,1,0,1,0] => [5,4,1,1,1] => 67

[1,0,1,1,1,0,0,0,1,1,0,0] => [4,4,1,1,1] => 14

[1,0,1,1,1,0,0,1,0,0,1,0] => [5,3,1,1,1] => 31

[1,0,1,1,1,0,0,1,0,1,0,0] => [4,3,1,1,1] => 11

[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] => 12

[1,0,1,1,1,0,1,0,0,1,0,0] => [4,2,1,1,1] => 6

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

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

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

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

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

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

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

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searching the database for the individual values of this statistic

Description

The largest multiplicity of an irreducible representation 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 $(n-1,\ldots,2,1)$.