- FindStat

Identifier

Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words

Images

[1] => [1] => 1 => 0

[1,1] => [2] => 10 => 01

[2] => [1,1] => 11 => 00

[1,1,1] => [3] => 100 => 011

[1,2] => [2,1] => 101 => 010

[2,1] => [1,2] => 110 => 001

[3] => [1,1,1] => 111 => 000

[1,1,1,1] => [4] => 1000 => 0111

[1,1,2] => [3,1] => 1001 => 0110

[1,2,1] => [2,2] => 1010 => 0101

[1,3] => [2,1,1] => 1011 => 0100

[2,1,1] => [1,3] => 1100 => 0011

[2,2] => [1,2,1] => 1101 => 0010

[3,1] => [1,1,2] => 1110 => 0001

[4] => [1,1,1,1] => 1111 => 0000

[1,1,1,1,1] => [5] => 10000 => 01111

[1,1,1,2] => [4,1] => 10001 => 01110

[1,1,2,1] => [3,2] => 10010 => 01101

[1,1,3] => [3,1,1] => 10011 => 01100

[1,2,1,1] => [2,3] => 10100 => 01011

[1,2,2] => [2,2,1] => 10101 => 01010

[1,3,1] => [2,1,2] => 10110 => 01001

[1,4] => [2,1,1,1] => 10111 => 01000

[2,1,1,1] => [1,4] => 11000 => 00111

[2,1,2] => [1,3,1] => 11001 => 00110

[2,2,1] => [1,2,2] => 11010 => 00101

[2,3] => [1,2,1,1] => 11011 => 00100

[3,1,1] => [1,1,3] => 11100 => 00011

[3,2] => [1,1,2,1] => 11101 => 00010

[4,1] => [1,1,1,2] => 11110 => 00001

[5] => [1,1,1,1,1] => 11111 => 00000

[1,1,1,1,1,1] => [6] => 100000 => 011111

[1,1,1,1,2] => [5,1] => 100001 => 011110

[1,1,1,2,1] => [4,2] => 100010 => 011101

[1,1,1,3] => [4,1,1] => 100011 => 011100

[1,1,2,1,1] => [3,3] => 100100 => 011011

[1,1,2,2] => [3,2,1] => 100101 => 011010

[1,1,3,1] => [3,1,2] => 100110 => 011001

[1,1,4] => [3,1,1,1] => 100111 => 011000

[1,2,1,1,1] => [2,4] => 101000 => 010111

[1,2,1,2] => [2,3,1] => 101001 => 010110

[1,2,2,1] => [2,2,2] => 101010 => 010101

[1,2,3] => [2,2,1,1] => 101011 => 010100

[1,3,1,1] => [2,1,3] => 101100 => 010011

[1,3,2] => [2,1,2,1] => 101101 => 010010

[1,4,1] => [2,1,1,2] => 101110 => 010001

[1,5] => [2,1,1,1,1] => 101111 => 010000

[2,1,1,1,1] => [1,5] => 110000 => 001111

[2,1,1,2] => [1,4,1] => 110001 => 001110

[2,1,2,1] => [1,3,2] => 110010 => 001101

[2,1,3] => [1,3,1,1] => 110011 => 001100

[2,2,1,1] => [1,2,3] => 110100 => 001011

[2,2,2] => [1,2,2,1] => 110101 => 001010

[2,3,1] => [1,2,1,2] => 110110 => 001001

[2,4] => [1,2,1,1,1] => 110111 => 001000

[3,1,1,1] => [1,1,4] => 111000 => 000111

[3,1,2] => [1,1,3,1] => 111001 => 000110

[3,2,1] => [1,1,2,2] => 111010 => 000101

[3,3] => [1,1,2,1,1] => 111011 => 000100

[4,1,1] => [1,1,1,3] => 111100 => 000011

[4,2] => [1,1,1,2,1] => 111101 => 000010

[5,1] => [1,1,1,1,2] => 111110 => 000001

[6] => [1,1,1,1,1,1] => 111111 => 000000

[1,1,1,1,1,1,1] => [7] => 1000000 => 0111111

[1,1,1,1,1,2] => [6,1] => 1000001 => 0111110

[1,1,1,1,2,1] => [5,2] => 1000010 => 0111101

[1,1,1,1,3] => [5,1,1] => 1000011 => 0111100

[1,1,1,2,1,1] => [4,3] => 1000100 => 0111011

[1,1,1,2,2] => [4,2,1] => 1000101 => 0111010

[1,1,1,3,1] => [4,1,2] => 1000110 => 0111001

[1,1,1,4] => [4,1,1,1] => 1000111 => 0111000

[1,1,2,1,1,1] => [3,4] => 1001000 => 0110111

[1,1,2,1,2] => [3,3,1] => 1001001 => 0110110

[1,1,2,2,1] => [3,2,2] => 1001010 => 0110101

[1,1,2,3] => [3,2,1,1] => 1001011 => 0110100

[1,1,3,1,1] => [3,1,3] => 1001100 => 0110011

[1,1,3,2] => [3,1,2,1] => 1001101 => 0110010

[1,1,4,1] => [3,1,1,2] => 1001110 => 0110001

[1,1,5] => [3,1,1,1,1] => 1001111 => 0110000

[1,2,1,1,1,1] => [2,5] => 1010000 => 0101111

[1,2,1,1,2] => [2,4,1] => 1010001 => 0101110

[1,2,1,2,1] => [2,3,2] => 1010010 => 0101101

[1,2,1,3] => [2,3,1,1] => 1010011 => 0101100

[1,2,2,1,1] => [2,2,3] => 1010100 => 0101011

[1,2,2,2] => [2,2,2,1] => 1010101 => 0101010

[1,2,3,1] => [2,2,1,2] => 1010110 => 0101001

[1,2,4] => [2,2,1,1,1] => 1010111 => 0101000

[1,3,1,1,1] => [2,1,4] => 1011000 => 0100111

[1,3,1,2] => [2,1,3,1] => 1011001 => 0100110

[1,3,2,1] => [2,1,2,2] => 1011010 => 0100101

[1,3,3] => [2,1,2,1,1] => 1011011 => 0100100

[1,4,1,1] => [2,1,1,3] => 1011100 => 0100011

[1,4,2] => [2,1,1,2,1] => 1011101 => 0100010

[1,5,1] => [2,1,1,1,2] => 1011110 => 0100001

[1,6] => [2,1,1,1,1,1] => 1011111 => 0100000

[2,1,1,1,1,1] => [1,6] => 1100000 => 0011111

[2,1,1,1,2] => [1,5,1] => 1100001 => 0011110

[2,1,1,2,1] => [1,4,2] => 1100010 => 0011101

[2,1,1,3] => [1,4,1,1] => 1100011 => 0011100

[2,1,2,1,1] => [1,3,3] => 1100100 => 0011011

[2,1,2,2] => [1,3,2,1] => 1100101 => 0011010

>>> Load all 1615 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 $0$.

Map

complement

Description

Send a binary word to the word obtained by interchanging the two letters.