- FindStat

Identifier

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000392: Binary words ⟶ ℤ

Values

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

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

00 => [2] => 10 => 01 => 1

01 => [1,1] => 11 => 00 => 0

10 => [1,1] => 11 => 00 => 0

11 => [2] => 10 => 01 => 1

000 => [3] => 100 => 011 => 2

001 => [2,1] => 101 => 010 => 1

010 => [1,1,1] => 111 => 000 => 0

011 => [1,2] => 110 => 001 => 1

100 => [1,2] => 110 => 001 => 1

101 => [1,1,1] => 111 => 000 => 0

110 => [2,1] => 101 => 010 => 1

111 => [3] => 100 => 011 => 2

0000 => [4] => 1000 => 0111 => 3

0001 => [3,1] => 1001 => 0110 => 2

0010 => [2,1,1] => 1011 => 0100 => 1

0011 => [2,2] => 1010 => 0101 => 1

0100 => [1,1,2] => 1110 => 0001 => 1

0101 => [1,1,1,1] => 1111 => 0000 => 0

0110 => [1,2,1] => 1101 => 0010 => 1

0111 => [1,3] => 1100 => 0011 => 2

1000 => [1,3] => 1100 => 0011 => 2

1001 => [1,2,1] => 1101 => 0010 => 1

1010 => [1,1,1,1] => 1111 => 0000 => 0

1011 => [1,1,2] => 1110 => 0001 => 1

1100 => [2,2] => 1010 => 0101 => 1

1101 => [2,1,1] => 1011 => 0100 => 1

1110 => [3,1] => 1001 => 0110 => 2

1111 => [4] => 1000 => 0111 => 3

00000 => [5] => 10000 => 01111 => 4

00001 => [4,1] => 10001 => 01110 => 3

00010 => [3,1,1] => 10011 => 01100 => 2

00011 => [3,2] => 10010 => 01101 => 2

00100 => [2,1,2] => 10110 => 01001 => 1

00101 => [2,1,1,1] => 10111 => 01000 => 1

00110 => [2,2,1] => 10101 => 01010 => 1

00111 => [2,3] => 10100 => 01011 => 2

01000 => [1,1,3] => 11100 => 00011 => 2

01001 => [1,1,2,1] => 11101 => 00010 => 1

01010 => [1,1,1,1,1] => 11111 => 00000 => 0

01011 => [1,1,1,2] => 11110 => 00001 => 1

01100 => [1,2,2] => 11010 => 00101 => 1

01101 => [1,2,1,1] => 11011 => 00100 => 1

01110 => [1,3,1] => 11001 => 00110 => 2

01111 => [1,4] => 11000 => 00111 => 3

10000 => [1,4] => 11000 => 00111 => 3

10001 => [1,3,1] => 11001 => 00110 => 2

10010 => [1,2,1,1] => 11011 => 00100 => 1

10011 => [1,2,2] => 11010 => 00101 => 1

10100 => [1,1,1,2] => 11110 => 00001 => 1

10101 => [1,1,1,1,1] => 11111 => 00000 => 0

10110 => [1,1,2,1] => 11101 => 00010 => 1

10111 => [1,1,3] => 11100 => 00011 => 2

11000 => [2,3] => 10100 => 01011 => 2

11001 => [2,2,1] => 10101 => 01010 => 1

11010 => [2,1,1,1] => 10111 => 01000 => 1

11011 => [2,1,2] => 10110 => 01001 => 1

11100 => [3,2] => 10010 => 01101 => 2

11101 => [3,1,1] => 10011 => 01100 => 2

11110 => [4,1] => 10001 => 01110 => 3

11111 => [5] => 10000 => 01111 => 4

000000 => [6] => 100000 => 011111 => 5

000001 => [5,1] => 100001 => 011110 => 4

000010 => [4,1,1] => 100011 => 011100 => 3

000011 => [4,2] => 100010 => 011101 => 3

000100 => [3,1,2] => 100110 => 011001 => 2

000101 => [3,1,1,1] => 100111 => 011000 => 2

000110 => [3,2,1] => 100101 => 011010 => 2

000111 => [3,3] => 100100 => 011011 => 2

001000 => [2,1,3] => 101100 => 010011 => 2

001001 => [2,1,2,1] => 101101 => 010010 => 1

001010 => [2,1,1,1,1] => 101111 => 010000 => 1

001011 => [2,1,1,2] => 101110 => 010001 => 1

001100 => [2,2,2] => 101010 => 010101 => 1

001101 => [2,2,1,1] => 101011 => 010100 => 1

001110 => [2,3,1] => 101001 => 010110 => 2

001111 => [2,4] => 101000 => 010111 => 3

010000 => [1,1,4] => 111000 => 000111 => 3

010001 => [1,1,3,1] => 111001 => 000110 => 2

010010 => [1,1,2,1,1] => 111011 => 000100 => 1

010011 => [1,1,2,2] => 111010 => 000101 => 1

010100 => [1,1,1,1,2] => 111110 => 000001 => 1

010101 => [1,1,1,1,1,1] => 111111 => 000000 => 0

010110 => [1,1,1,2,1] => 111101 => 000010 => 1

010111 => [1,1,1,3] => 111100 => 000011 => 2

011000 => [1,2,3] => 110100 => 001011 => 2

011001 => [1,2,2,1] => 110101 => 001010 => 1

011010 => [1,2,1,1,1] => 110111 => 001000 => 1

011011 => [1,2,1,2] => 110110 => 001001 => 1

011100 => [1,3,2] => 110010 => 001101 => 2

011101 => [1,3,1,1] => 110011 => 001100 => 2

011110 => [1,4,1] => 110001 => 001110 => 3

011111 => [1,5] => 110000 => 001111 => 4

100000 => [1,5] => 110000 => 001111 => 4

100001 => [1,4,1] => 110001 => 001110 => 3

100010 => [1,3,1,1] => 110011 => 001100 => 2

100011 => [1,3,2] => 110010 => 001101 => 2

100100 => [1,2,1,2] => 110110 => 001001 => 1

100101 => [1,2,1,1,1] => 110111 => 001000 => 1

100110 => [1,2,2,1] => 110101 => 001010 => 1

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Description

The length of the longest run of ones in a binary word.

Map

delta morphism

Description

Applies the delta morphism to a binary word.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.

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.