- FindStat

Identifier

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000382: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The first part of an integer composition.

Map

reverse

Description

Return the reversal of a composition.
That is, the composition $(i_1, i_2, \ldots, i_k)$ is sent to $(i_k, i_{k-1}, \ldots, i_1)$.

Map

rise composition

Description

Send a Dyck path to the composition of sizes of its rises.