- FindStat

Identifier

Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words

Images

[1] => [1] => 1

[1,1] => [2] => 10

[2] => [1,1] => 11

[1,1,1] => [3] => 100

[1,2] => [2,1] => 101

[2,1] => [1,2] => 110

[3] => [1,1,1] => 111

[1,1,1,1] => [4] => 1000

[1,1,2] => [3,1] => 1001

[1,2,1] => [2,2] => 1010

[1,3] => [2,1,1] => 1011

[2,1,1] => [1,3] => 1100

[2,2] => [1,2,1] => 1101

[3,1] => [1,1,2] => 1110

[4] => [1,1,1,1] => 1111

[1,1,1,1,1] => [5] => 10000

[1,1,1,2] => [4,1] => 10001

[1,1,2,1] => [3,2] => 10010

[1,1,3] => [3,1,1] => 10011

[1,2,1,1] => [2,3] => 10100

[1,2,2] => [2,2,1] => 10101

[1,3,1] => [2,1,2] => 10110

[1,4] => [2,1,1,1] => 10111

[2,1,1,1] => [1,4] => 11000

[2,1,2] => [1,3,1] => 11001

[2,2,1] => [1,2,2] => 11010

[2,3] => [1,2,1,1] => 11011

[3,1,1] => [1,1,3] => 11100

[3,2] => [1,1,2,1] => 11101

[4,1] => [1,1,1,2] => 11110

[5] => [1,1,1,1,1] => 11111

[1,1,1,1,1,1] => [6] => 100000

[1,1,1,1,2] => [5,1] => 100001

[1,1,1,2,1] => [4,2] => 100010

[1,1,1,3] => [4,1,1] => 100011

[1,1,2,1,1] => [3,3] => 100100

[1,1,2,2] => [3,2,1] => 100101

[1,1,3,1] => [3,1,2] => 100110

[1,1,4] => [3,1,1,1] => 100111

[1,2,1,1,1] => [2,4] => 101000

[1,2,1,2] => [2,3,1] => 101001

[1,2,2,1] => [2,2,2] => 101010

[1,2,3] => [2,2,1,1] => 101011

[1,3,1,1] => [2,1,3] => 101100

[1,3,2] => [2,1,2,1] => 101101

[1,4,1] => [2,1,1,2] => 101110

[1,5] => [2,1,1,1,1] => 101111

[2,1,1,1,1] => [1,5] => 110000

[2,1,1,2] => [1,4,1] => 110001

[2,1,2,1] => [1,3,2] => 110010

[2,1,3] => [1,3,1,1] => 110011

[2,2,1,1] => [1,2,3] => 110100

[2,2,2] => [1,2,2,1] => 110101

[2,3,1] => [1,2,1,2] => 110110

[2,4] => [1,2,1,1,1] => 110111

[3,1,1,1] => [1,1,4] => 111000

[3,1,2] => [1,1,3,1] => 111001

[3,2,1] => [1,1,2,2] => 111010

[3,3] => [1,1,2,1,1] => 111011

[4,1,1] => [1,1,1,3] => 111100

[4,2] => [1,1,1,2,1] => 111101

[5,1] => [1,1,1,1,2] => 111110

[6] => [1,1,1,1,1,1] => 111111

[1,1,1,1,1,1,1] => [7] => 1000000

[1,1,1,1,1,2] => [6,1] => 1000001

[1,1,1,1,2,1] => [5,2] => 1000010

[1,1,1,1,3] => [5,1,1] => 1000011

[1,1,1,2,1,1] => [4,3] => 1000100

[1,1,1,2,2] => [4,2,1] => 1000101

[1,1,1,3,1] => [4,1,2] => 1000110

[1,1,1,4] => [4,1,1,1] => 1000111

[1,1,2,1,1,1] => [3,4] => 1001000

[1,1,2,1,2] => [3,3,1] => 1001001

[1,1,2,2,1] => [3,2,2] => 1001010

[1,1,2,3] => [3,2,1,1] => 1001011

[1,1,3,1,1] => [3,1,3] => 1001100

[1,1,3,2] => [3,1,2,1] => 1001101

[1,1,4,1] => [3,1,1,2] => 1001110

[1,1,5] => [3,1,1,1,1] => 1001111

[1,2,1,1,1,1] => [2,5] => 1010000

[1,2,1,1,2] => [2,4,1] => 1010001

[1,2,1,2,1] => [2,3,2] => 1010010

[1,2,1,3] => [2,3,1,1] => 1010011

[1,2,2,1,1] => [2,2,3] => 1010100

[1,2,2,2] => [2,2,2,1] => 1010101

[1,2,3,1] => [2,2,1,2] => 1010110

[1,2,4] => [2,2,1,1,1] => 1010111

[1,3,1,1,1] => [2,1,4] => 1011000

[1,3,1,2] => [2,1,3,1] => 1011001

[1,3,2,1] => [2,1,2,2] => 1011010

[1,3,3] => [2,1,2,1,1] => 1011011

[1,4,1,1] => [2,1,1,3] => 1011100

[1,4,2] => [2,1,1,2,1] => 1011101

[1,5,1] => [2,1,1,1,2] => 1011110

[1,6] => [2,1,1,1,1,1] => 1011111

[2,1,1,1,1,1] => [1,6] => 1100000

[2,1,1,1,2] => [1,5,1] => 1100001

[2,1,1,2,1] => [1,4,2] => 1100010

[2,1,1,3] => [1,4,1,1] => 1100011

[2,1,2,1,1] => [1,3,3] => 1100100

[2,1,2,2] => [1,3,2,1] => 1100101

>>> Load all 1687 entries. <<<

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 composition as a binary word, treating ones as separators.
Encoding a positive integer $i$ as the word $10\dots 0$ consisting of a one followed by $i-1$ zeros, the binary word of a composition $(i_1,\dots,i_k)$ is the concatenation of of words for $i_1,\dots,i_k$.
The image of this map contains precisely the words which do not begin with a $0$.