Identifier
-
Mp00039:
Integer compositions
—complement⟶
Integer compositions
St000008: Integer compositions ⟶ ℤ
Values
[1] => [1] => 0
[1,1] => [2] => 0
[2] => [1,1] => 1
[1,1,1] => [3] => 0
[1,2] => [2,1] => 2
[2,1] => [1,2] => 1
[3] => [1,1,1] => 3
[1,1,1,1] => [4] => 0
[1,1,2] => [3,1] => 3
[1,2,1] => [2,2] => 2
[1,3] => [2,1,1] => 5
[2,1,1] => [1,3] => 1
[2,2] => [1,2,1] => 4
[3,1] => [1,1,2] => 3
[4] => [1,1,1,1] => 6
[1,1,1,1,1] => [5] => 0
[1,1,1,2] => [4,1] => 4
[1,1,2,1] => [3,2] => 3
[1,1,3] => [3,1,1] => 7
[1,2,1,1] => [2,3] => 2
[1,2,2] => [2,2,1] => 6
[1,3,1] => [2,1,2] => 5
[1,4] => [2,1,1,1] => 9
[2,1,1,1] => [1,4] => 1
[2,1,2] => [1,3,1] => 5
[2,2,1] => [1,2,2] => 4
[2,3] => [1,2,1,1] => 8
[3,1,1] => [1,1,3] => 3
[3,2] => [1,1,2,1] => 7
[4,1] => [1,1,1,2] => 6
[5] => [1,1,1,1,1] => 10
[1,1,1,1,1,1] => [6] => 0
[1,1,1,1,2] => [5,1] => 5
[1,1,1,2,1] => [4,2] => 4
[1,1,1,3] => [4,1,1] => 9
[1,1,2,1,1] => [3,3] => 3
[1,1,2,2] => [3,2,1] => 8
[1,1,3,1] => [3,1,2] => 7
[1,1,4] => [3,1,1,1] => 12
[1,2,1,1,1] => [2,4] => 2
[1,2,1,2] => [2,3,1] => 7
[1,2,2,1] => [2,2,2] => 6
[1,2,3] => [2,2,1,1] => 11
[1,3,1,1] => [2,1,3] => 5
[1,3,2] => [2,1,2,1] => 10
[1,4,1] => [2,1,1,2] => 9
[1,5] => [2,1,1,1,1] => 14
[2,1,1,1,1] => [1,5] => 1
[2,1,1,2] => [1,4,1] => 6
[2,1,2,1] => [1,3,2] => 5
[2,1,3] => [1,3,1,1] => 10
[2,2,1,1] => [1,2,3] => 4
[2,2,2] => [1,2,2,1] => 9
[2,3,1] => [1,2,1,2] => 8
[2,4] => [1,2,1,1,1] => 13
[3,1,1,1] => [1,1,4] => 3
[3,1,2] => [1,1,3,1] => 8
[3,2,1] => [1,1,2,2] => 7
[3,3] => [1,1,2,1,1] => 12
[4,1,1] => [1,1,1,3] => 6
[4,2] => [1,1,1,2,1] => 11
[5,1] => [1,1,1,1,2] => 10
[6] => [1,1,1,1,1,1] => 15
[1,1,1,1,1,1,1] => [7] => 0
[1,1,1,1,1,2] => [6,1] => 6
[1,1,1,1,2,1] => [5,2] => 5
[1,1,1,1,3] => [5,1,1] => 11
[1,1,1,2,1,1] => [4,3] => 4
[1,1,1,2,2] => [4,2,1] => 10
[1,1,1,3,1] => [4,1,2] => 9
[1,1,1,4] => [4,1,1,1] => 15
[1,1,2,1,1,1] => [3,4] => 3
[1,1,2,1,2] => [3,3,1] => 9
[1,1,2,2,1] => [3,2,2] => 8
[1,1,2,3] => [3,2,1,1] => 14
[1,1,3,1,1] => [3,1,3] => 7
[1,1,3,2] => [3,1,2,1] => 13
[1,1,4,1] => [3,1,1,2] => 12
[1,1,5] => [3,1,1,1,1] => 18
[1,2,1,1,1,1] => [2,5] => 2
[1,2,1,1,2] => [2,4,1] => 8
[1,2,1,2,1] => [2,3,2] => 7
[1,2,1,3] => [2,3,1,1] => 13
[1,2,2,1,1] => [2,2,3] => 6
[1,2,2,2] => [2,2,2,1] => 12
[1,2,3,1] => [2,2,1,2] => 11
[1,2,4] => [2,2,1,1,1] => 17
[1,3,1,1,1] => [2,1,4] => 5
[1,3,1,2] => [2,1,3,1] => 11
[1,3,2,1] => [2,1,2,2] => 10
[1,3,3] => [2,1,2,1,1] => 16
[1,4,1,1] => [2,1,1,3] => 9
[1,4,2] => [2,1,1,2,1] => 15
[1,5,1] => [2,1,1,1,2] => 14
[1,6] => [2,1,1,1,1,1] => 20
[2,1,1,1,1,1] => [1,6] => 1
[2,1,1,1,2] => [1,5,1] => 7
[2,1,1,2,1] => [1,4,2] => 6
[2,1,1,3] => [1,4,1,1] => 12
[2,1,2,1,1] => [1,3,3] => 5
[2,1,2,2] => [1,3,2,1] => 11
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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
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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