- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words

Images

0 => [2] => 10 => 11

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

00 => [3] => 100 => 101

01 => [2,1] => 101 => 110

10 => [1,2] => 110 => 111

11 => [1,1,1] => 111 => 111

000 => [4] => 1000 => 1001

001 => [3,1] => 1001 => 1010

010 => [2,2] => 1010 => 1101

011 => [2,1,1] => 1011 => 1101

100 => [1,3] => 1100 => 1101

101 => [1,2,1] => 1101 => 1110

110 => [1,1,2] => 1110 => 1111

111 => [1,1,1,1] => 1111 => 1111

0000 => [5] => 10000 => 10001

0001 => [4,1] => 10001 => 10010

0010 => [3,2] => 10010 => 10101

0011 => [3,1,1] => 10011 => 10101

0100 => [2,3] => 10100 => 11001

0101 => [2,2,1] => 10101 => 11010

0110 => [2,1,2] => 10110 => 11011

0111 => [2,1,1,1] => 10111 => 11011

1000 => [1,4] => 11000 => 11001

1001 => [1,3,1] => 11001 => 11010

1010 => [1,2,2] => 11010 => 11101

1011 => [1,2,1,1] => 11011 => 11101

1100 => [1,1,3] => 11100 => 11101

1101 => [1,1,2,1] => 11101 => 11110

1110 => [1,1,1,2] => 11110 => 11111

1111 => [1,1,1,1,1] => 11111 => 11111

00000 => [6] => 100000 => 100001

00001 => [5,1] => 100001 => 100010

00010 => [4,2] => 100010 => 100101

00011 => [4,1,1] => 100011 => 100101

00100 => [3,3] => 100100 => 101001

00101 => [3,2,1] => 100101 => 101010

00110 => [3,1,2] => 100110 => 101011

00111 => [3,1,1,1] => 100111 => 101011

01000 => [2,4] => 101000 => 110001

01001 => [2,3,1] => 101001 => 110010

01010 => [2,2,2] => 101010 => 110101

01011 => [2,2,1,1] => 101011 => 110101

01100 => [2,1,3] => 101100 => 110101

01101 => [2,1,2,1] => 101101 => 110110

01110 => [2,1,1,2] => 101110 => 110111

01111 => [2,1,1,1,1] => 101111 => 110111

10000 => [1,5] => 110000 => 110001

10001 => [1,4,1] => 110001 => 110010

10010 => [1,3,2] => 110010 => 110101

10011 => [1,3,1,1] => 110011 => 110101

10100 => [1,2,3] => 110100 => 111001

10101 => [1,2,2,1] => 110101 => 111010

10110 => [1,2,1,2] => 110110 => 111011

10111 => [1,2,1,1,1] => 110111 => 111011

11000 => [1,1,4] => 111000 => 111001

11001 => [1,1,3,1] => 111001 => 111010

11010 => [1,1,2,2] => 111010 => 111101

11011 => [1,1,2,1,1] => 111011 => 111101

11100 => [1,1,1,3] => 111100 => 111101

11101 => [1,1,1,2,1] => 111101 => 111110

11110 => [1,1,1,1,2] => 111110 => 111111

11111 => [1,1,1,1,1,1] => 111111 => 111111

000000 => [7] => 1000000 => 1000001

000001 => [6,1] => 1000001 => 1000010

000010 => [5,2] => 1000010 => 1000101

000011 => [5,1,1] => 1000011 => 1000101

000100 => [4,3] => 1000100 => 1001001

000101 => [4,2,1] => 1000101 => 1001010

000110 => [4,1,2] => 1000110 => 1001011

000111 => [4,1,1,1] => 1000111 => 1001011

001000 => [3,4] => 1001000 => 1010001

001001 => [3,3,1] => 1001001 => 1010010

001010 => [3,2,2] => 1001010 => 1010101

001011 => [3,2,1,1] => 1001011 => 1010101

001100 => [3,1,3] => 1001100 => 1010101

001101 => [3,1,2,1] => 1001101 => 1010110

001110 => [3,1,1,2] => 1001110 => 1010111

001111 => [3,1,1,1,1] => 1001111 => 1010111

010000 => [2,5] => 1010000 => 1100001

010001 => [2,4,1] => 1010001 => 1100010

010010 => [2,3,2] => 1010010 => 1100101

010011 => [2,3,1,1] => 1010011 => 1100101

010100 => [2,2,3] => 1010100 => 1101001

010101 => [2,2,2,1] => 1010101 => 1101010

010110 => [2,2,1,2] => 1010110 => 1101011

010111 => [2,2,1,1,1] => 1010111 => 1101011

011000 => [2,1,4] => 1011000 => 1101001

011001 => [2,1,3,1] => 1011001 => 1101010

011010 => [2,1,2,2] => 1011010 => 1101101

011011 => [2,1,2,1,1] => 1011011 => 1101101

011100 => [2,1,1,3] => 1011100 => 1101101

011101 => [2,1,1,2,1] => 1011101 => 1101110

011110 => [2,1,1,1,2] => 1011110 => 1101111

011111 => [2,1,1,1,1,1] => 1011111 => 1101111

100000 => [1,6] => 1100000 => 1100001

100001 => [1,5,1] => 1100001 => 1100010

100010 => [1,4,2] => 1100010 => 1100101

100011 => [1,4,1,1] => 1100011 => 1100101

100100 => [1,3,3] => 1100100 => 1101001

100101 => [1,3,2,1] => 1100101 => 1101010

100110 => [1,3,1,2] => 1100110 => 1101011

>>> Load all 737 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

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

valleys-to-peaks

Description

Return the binary word with every valley replaced by a peak.
A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. This map replaces every valley with a peak.