- FindStat

Identifier

Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001481: Dyck paths ⟶ ℤ

Values

[1] => [1] => [1,0] => 1
[1,1] => [2] => [1,1,0,0] => 2
[2] => [1,1] => [1,0,1,0] => 1
[1,1,1] => [3] => [1,1,1,0,0,0] => 3
[1,2] => [2,1] => [1,1,0,0,1,0] => 1
[2,1] => [1,2] => [1,0,1,1,0,0] => 1
[3] => [1,1,1] => [1,0,1,0,1,0] => 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0] => 4
[1,1,2] => [3,1] => [1,1,1,0,0,0,1,0] => 1
[1,2,1] => [2,2] => [1,1,0,0,1,1,0,0] => 2
[1,3] => [2,1,1] => [1,1,0,0,1,0,1,0] => 1
[2,1,1] => [1,3] => [1,0,1,1,1,0,0,0] => 1
[2,2] => [1,2,1] => [1,0,1,1,0,0,1,0] => 1
[3,1] => [1,1,2] => [1,0,1,0,1,1,0,0] => 1
[4] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0] => 5
[1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 1
[1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
[1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 1
[1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2
[1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 1
[1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 1
[1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 1
[2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1
[2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 1
[2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 1
[2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 1
[3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 1
[3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 1
[4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 1
[5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 6
[1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[1,1,1,2,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
[1,1,1,3] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 1
[1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
[1,1,2,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 1
[1,1,3,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 1
[1,1,4] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 1
[1,2,1,1,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2
[1,2,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 1
[1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 2
[1,2,3] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 1
[1,3,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 1
[1,3,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 1
[1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 1
[1,5] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 1
[2,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
[2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 1
[2,1,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 1
[2,1,3] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 1
[2,2,1,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 1
[2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 1
[2,3,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 1
[2,4] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 1
[3,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 1
[3,1,2] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 1
[3,2,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 1
[3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 1
[4,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 1
[4,2] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 1
[5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 1
[6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1

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$F_{1} = q$

$F_{2} = q + q^{2}$

$F_{3} = 3\ q + q^{3}$

$F_{4} = 6\ q + q^{2} + q^{4}$

$F_{5} = 13\ q + 2\ q^{2} + q^{5}$

$F_{6} = 27\ q + 3\ q^{2} + q^{3} + q^{6}$

Description

The minimal height of a peak of a Dyck path.

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

$I$ .

Map

bounce path

Description

The bounce path determined by an integer composition.