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Identifier

Mp00305: Permutations —parking function⟶ Parking functions
Mp00319: Parking functions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000008: Integer compositions ⟶ ℤ

Values

[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [2,1] => 2
[2,1] => [2,1] => [2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [2,2,1,1] => 11
[1,3,2] => [1,3,2] => [1,3,2] => [2,1,2,1] => 10
[2,1,3] => [2,1,3] => [2,1,3] => [1,3,1,1] => 10
[2,3,1] => [2,3,1] => [2,3,1] => [1,2,1,2] => 8
[3,1,2] => [3,1,2] => [3,1,2] => [1,1,3,1] => 8
[3,2,1] => [3,2,1] => [3,2,1] => [1,1,2,2] => 7

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$F_{1} = 1$

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

$F_{3} = q^{7} + 2\ q^{8} + 2\ q^{10} + q^{11}$

Description

The major index of the composition.
The descents of a composition

$[c_1,c_2,\dots,c_k]$ are the partial sums

$c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$ , excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see Permutations/Descents-Major.

Map

parking function

Description

Interpret the permutation as a parking function.

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

to composition

Description

Return the parking function interpreted as an integer composition.