- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions

Images

0 => [2] => [1,1]

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

00 => [3] => [1,1,1]

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

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

11 => [1,1,1] => [3]

000 => [4] => [1,1,1,1]

001 => [3,1] => [1,1,2]

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

011 => [2,1,1] => [1,3]

100 => [1,3] => [2,1,1]

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

110 => [1,1,2] => [3,1]

111 => [1,1,1,1] => [4]

0000 => [5] => [1,1,1,1,1]

0001 => [4,1] => [1,1,1,2]

0010 => [3,2] => [1,1,2,1]

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

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

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

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

0111 => [2,1,1,1] => [1,4]

1000 => [1,4] => [2,1,1,1]

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

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

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

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

1101 => [1,1,2,1] => [3,2]

1110 => [1,1,1,2] => [4,1]

1111 => [1,1,1,1,1] => [5]

00000 => [6] => [1,1,1,1,1,1]

00001 => [5,1] => [1,1,1,1,2]

00010 => [4,2] => [1,1,1,2,1]

00011 => [4,1,1] => [1,1,1,3]

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

00101 => [3,2,1] => [1,1,2,2]

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

00111 => [3,1,1,1] => [1,1,4]

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

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

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

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

01100 => [2,1,3] => [1,3,1,1]

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

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

01111 => [2,1,1,1,1] => [1,5]

10000 => [1,5] => [2,1,1,1,1]

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

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

10011 => [1,3,1,1] => [2,1,3]

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

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

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

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

11000 => [1,1,4] => [3,1,1,1]

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

11010 => [1,1,2,2] => [3,2,1]

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

11100 => [1,1,1,3] => [4,1,1]

11101 => [1,1,1,2,1] => [4,2]

11110 => [1,1,1,1,2] => [5,1]

11111 => [1,1,1,1,1,1] => [6]

000000 => [7] => [1,1,1,1,1,1,1]

000001 => [6,1] => [1,1,1,1,1,2]

000010 => [5,2] => [1,1,1,1,2,1]

000011 => [5,1,1] => [1,1,1,1,3]

000100 => [4,3] => [1,1,1,2,1,1]

000101 => [4,2,1] => [1,1,1,2,2]

000110 => [4,1,2] => [1,1,1,3,1]

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

001000 => [3,4] => [1,1,2,1,1,1]

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

001010 => [3,2,2] => [1,1,2,2,1]

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

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

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

001110 => [3,1,1,2] => [1,1,4,1]

001111 => [3,1,1,1,1] => [1,1,5]

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

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

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

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

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

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

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

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

011000 => [2,1,4] => [1,3,1,1,1]

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

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

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

011100 => [2,1,1,3] => [1,4,1,1]

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

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

011111 => [2,1,1,1,1,1] => [1,6]

100000 => [1,6] => [2,1,1,1,1,1]

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

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

100011 => [1,4,1,1] => [2,1,1,3]

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

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

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

>>> Load all 1914 entries. <<<

Map

to composition

Description

The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.

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 $I$.