Identifier
Values
[1,0] => 10 => [1,1] => [2] => 0
[1,0,1,0] => 1010 => [1,1,1,1] => [4] => 0
[1,1,0,0] => 1100 => [2,2] => [1,2,1] => 4
[1,0,1,0,1,0] => 101010 => [1,1,1,1,1,1] => [6] => 0
[1,0,1,1,0,0] => 101100 => [1,1,2,2] => [3,2,1] => 8
[1,1,0,0,1,0] => 110010 => [2,2,1,1] => [1,2,3] => 4
[1,1,0,1,0,0] => 110100 => [2,1,1,2] => [1,4,1] => 6
[1,1,1,0,0,0] => 111000 => [3,3] => [1,1,2,1,1] => 12
[1,0,1,0,1,0,1,0] => 10101010 => [1,1,1,1,1,1,1,1] => [8] => 0
[1,0,1,0,1,1,0,0] => 10101100 => [1,1,1,1,2,2] => [5,2,1] => 12
[1,0,1,1,0,0,1,0] => 10110010 => [1,1,2,2,1,1] => [3,2,3] => 8
[1,0,1,1,0,1,0,0] => 10110100 => [1,1,2,1,1,2] => [3,4,1] => 10
[1,0,1,1,1,0,0,0] => 10111000 => [1,1,3,3] => [3,1,2,1,1] => 20
[1,1,0,0,1,0,1,0] => 11001010 => [2,2,1,1,1,1] => [1,2,5] => 4
[1,1,0,0,1,1,0,0] => 11001100 => [2,2,2,2] => [1,2,2,2,1] => 16
[1,1,0,1,0,0,1,0] => 11010010 => [2,1,1,2,1,1] => [1,4,3] => 6
[1,1,0,1,0,1,0,0] => 11010100 => [2,1,1,1,1,2] => [1,6,1] => 8
[1,1,0,1,1,0,0,0] => 11011000 => [2,1,2,3] => [1,3,2,1,1] => 18
[1,1,1,0,0,0,1,0] => 11100010 => [3,3,1,1] => [1,1,2,1,3] => 12
[1,1,1,0,0,1,0,0] => 11100100 => [3,2,1,2] => [1,1,2,3,1] => 14
[1,1,1,0,1,0,0,0] => 11101000 => [3,1,1,3] => [1,1,4,1,1] => 16
[1,1,1,1,0,0,0,0] => 11110000 => [4,4] => [1,1,1,2,1,1,1] => 24
[1,1,0,1,0,1,0,1,0,0] => 1101010100 => [2,1,1,1,1,1,1,2] => [1,8,1] => 10
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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.
Map
delta morphism
Description
Applies the delta morphism to a binary word.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.
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 binary word
Description
Return the Dyck word as binary word.