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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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.
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
to composition
Description
Return the parking function interpreted as an integer composition.
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 $I$.
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 $I$.
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