- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤ

Values

0 => [2] => 10 => 1

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

00 => [3] => 100 => 1

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

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

11 => [1,1,1] => 111 => 3

000 => [4] => 1000 => 1

001 => [3,1] => 1001 => 2

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

011 => [2,1,1] => 1011 => 3

100 => [1,3] => 1100 => 2

101 => [1,2,1] => 1101 => 3

110 => [1,1,2] => 1110 => 3

111 => [1,1,1,1] => 1111 => 4

0000 => [5] => 10000 => 1

0001 => [4,1] => 10001 => 2

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

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

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

0101 => [2,2,1] => 10101 => 3

0110 => [2,1,2] => 10110 => 3

0111 => [2,1,1,1] => 10111 => 4

1000 => [1,4] => 11000 => 2

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

1010 => [1,2,2] => 11010 => 3

1011 => [1,2,1,1] => 11011 => 4

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

1101 => [1,1,2,1] => 11101 => 4

1110 => [1,1,1,2] => 11110 => 4

1111 => [1,1,1,1,1] => 11111 => 5

00000 => [6] => 100000 => 1

00001 => [5,1] => 100001 => 2

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

00011 => [4,1,1] => 100011 => 3

00100 => [3,3] => 100100 => 2

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

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

00111 => [3,1,1,1] => 100111 => 4

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

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

01010 => [2,2,2] => 101010 => 3

01011 => [2,2,1,1] => 101011 => 4

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

01101 => [2,1,2,1] => 101101 => 4

01110 => [2,1,1,2] => 101110 => 4

01111 => [2,1,1,1,1] => 101111 => 5

10000 => [1,5] => 110000 => 2

10001 => [1,4,1] => 110001 => 3

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

10011 => [1,3,1,1] => 110011 => 4

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

10101 => [1,2,2,1] => 110101 => 4

10110 => [1,2,1,2] => 110110 => 4

10111 => [1,2,1,1,1] => 110111 => 5

11000 => [1,1,4] => 111000 => 3

11001 => [1,1,3,1] => 111001 => 4

11010 => [1,1,2,2] => 111010 => 4

11011 => [1,1,2,1,1] => 111011 => 5

11100 => [1,1,1,3] => 111100 => 4

11101 => [1,1,1,2,1] => 111101 => 5

11110 => [1,1,1,1,2] => 111110 => 5

11111 => [1,1,1,1,1,1] => 111111 => 6

000000 => [7] => 1000000 => 1

000001 => [6,1] => 1000001 => 2

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

000011 => [5,1,1] => 1000011 => 3

000100 => [4,3] => 1000100 => 2

000101 => [4,2,1] => 1000101 => 3

000110 => [4,1,2] => 1000110 => 3

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

001000 => [3,4] => 1001000 => 2

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

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

001011 => [3,2,1,1] => 1001011 => 4

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

001101 => [3,1,2,1] => 1001101 => 4

001110 => [3,1,1,2] => 1001110 => 4

001111 => [3,1,1,1,1] => 1001111 => 5

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

010001 => [2,4,1] => 1010001 => 3

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

010011 => [2,3,1,1] => 1010011 => 4

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

010101 => [2,2,2,1] => 1010101 => 4

010110 => [2,2,1,2] => 1010110 => 4

010111 => [2,2,1,1,1] => 1010111 => 5

011000 => [2,1,4] => 1011000 => 3

011001 => [2,1,3,1] => 1011001 => 4

011010 => [2,1,2,2] => 1011010 => 4

011011 => [2,1,2,1,1] => 1011011 => 5

011100 => [2,1,1,3] => 1011100 => 4

011101 => [2,1,1,2,1] => 1011101 => 5

011110 => [2,1,1,1,2] => 1011110 => 5

011111 => [2,1,1,1,1,1] => 1011111 => 6

100000 => [1,6] => 1100000 => 2

100001 => [1,5,1] => 1100001 => 3

100010 => [1,4,2] => 1100010 => 3

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

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

100101 => [1,3,2,1] => 1100101 => 4

100110 => [1,3,1,2] => 1100110 => 4

>>> Load all 492 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The number of ones in a binary word.
This is also known as the Hamming weight of the word.

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 $0$.