- FindStat

Identifier

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Map

rise composition

Description

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

Map

complement

Description

The complement of a composition.
The complement of a composition $I$ is defined as follows:
If $I$ is the empty composition, then the complement is also the empty composition. Otherwise, let $S$ be the descent set corresponding to $I=(i_1,\dots,i_k)$, that is, the subset
$$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$$
of $\{ 1, 2, \ldots, |I|-1 \}$. Then, the complement of $I$ is the composition of the same size as $I$, whose descent set is $\{ 1, 2, \ldots, |I|-1 \} \setminus S$.
The complement of a composition $I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to $I$.