- FindStat

Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000766: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of inversions of an integer composition.
This is the number of pairs $(i,j)$ such that $i < j$ and $c_i > c_j$.

Map

touch composition

Description

Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.

Map

rotate front to back

Description

The front to back rotation of the entries of an integer composition.