- FindStat

Identifier

Mp00105: Binary words —complement⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions

Images

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

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

00 => 11 => 11 => [2]

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

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

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

000 => 111 => 111 => [3]

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

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

011 => 100 => 101 => [1,1,1]

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

101 => 010 => 100 => [1,2]

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

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

0000 => 1111 => 1111 => [4]

0001 => 1110 => 0111 => [1,3]

0010 => 1101 => 0011 => [2,2]

0011 => 1100 => 1011 => [1,1,2]

0100 => 1011 => 0001 => [3,1]

0101 => 1010 => 1001 => [1,2,1]

0110 => 1001 => 1101 => [2,1,1]

0111 => 1000 => 0101 => [1,1,1,1]

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

1001 => 0110 => 1000 => [1,3]

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

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

1100 => 0011 => 1110 => [3,1]

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

1110 => 0001 => 0010 => [2,1,1]

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

00000 => 11111 => 11111 => [5]

00001 => 11110 => 01111 => [1,4]

00010 => 11101 => 00111 => [2,3]

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

00100 => 11011 => 00011 => [3,2]

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

00110 => 11001 => 11011 => [2,1,2]

00111 => 11000 => 01011 => [1,1,1,2]

01000 => 10111 => 00001 => [4,1]

01001 => 10110 => 10001 => [1,3,1]

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

01011 => 10100 => 01001 => [1,1,2,1]

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

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

01110 => 10001 => 00101 => [2,1,1,1]

01111 => 10000 => 10101 => [1,1,1,1,1]

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

10001 => 01110 => 10000 => [1,4]

10010 => 01101 => 11000 => [2,3]

10011 => 01100 => 01000 => [1,1,3]

10100 => 01011 => 11100 => [3,2]

10101 => 01010 => 01100 => [1,2,2]

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

10111 => 01000 => 10100 => [1,1,1,2]

11000 => 00111 => 11110 => [4,1]

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

11010 => 00101 => 00110 => [2,2,1]

11011 => 00100 => 10110 => [1,1,2,1]

11100 => 00011 => 00010 => [3,1,1]

11101 => 00010 => 10010 => [1,2,1,1]

11110 => 00001 => 11010 => [2,1,1,1]

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

000000 => 111111 => 111111 => [6]

000001 => 111110 => 011111 => [1,5]

000010 => 111101 => 001111 => [2,4]

000011 => 111100 => 101111 => [1,1,4]

000100 => 111011 => 000111 => [3,3]

000101 => 111010 => 100111 => [1,2,3]

000110 => 111001 => 110111 => [2,1,3]

000111 => 111000 => 010111 => [1,1,1,3]

001000 => 110111 => 000011 => [4,2]

001001 => 110110 => 100011 => [1,3,2]

001010 => 110101 => 110011 => [2,2,2]

001011 => 110100 => 010011 => [1,1,2,2]

001100 => 110011 => 111011 => [3,1,2]

001101 => 110010 => 011011 => [1,2,1,2]

001110 => 110001 => 001011 => [2,1,1,2]

001111 => 110000 => 101011 => [1,1,1,1,2]

010000 => 101111 => 000001 => [5,1]

010001 => 101110 => 100001 => [1,4,1]

010010 => 101101 => 110001 => [2,3,1]

010011 => 101100 => 010001 => [1,1,3,1]

010100 => 101011 => 111001 => [3,2,1]

010101 => 101010 => 011001 => [1,2,2,1]

010110 => 101001 => 001001 => [2,1,2,1]

010111 => 101000 => 101001 => [1,1,1,2,1]

011000 => 100111 => 111101 => [4,1,1]

011001 => 100110 => 011101 => [1,3,1,1]

011010 => 100101 => 001101 => [2,2,1,1]

011011 => 100100 => 101101 => [1,1,2,1,1]

011100 => 100011 => 000101 => [3,1,1,1]

011101 => 100010 => 100101 => [1,2,1,1,1]

011110 => 100001 => 110101 => [2,1,1,1,1]

011111 => 100000 => 010101 => [1,1,1,1,1,1]

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

100001 => 011110 => 100000 => [1,5]

100010 => 011101 => 110000 => [2,4]

100011 => 011100 => 010000 => [1,1,4]

100100 => 011011 => 111000 => [3,3]

100101 => 011010 => 011000 => [1,2,3]

100110 => 011001 => 001000 => [2,1,3]

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Map

complement

Description

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

Map

zeros to flag zeros

Description

Return a binary word of the same length, such that the number of occurrences of $10$ in the word obtained by prepending the reverse of the complement equals the number of $0$s in the original word.
For example, the image of the word $w=1\dots1$ is $1\dots1$, because $w$ has no zeros, and $1\dots1$ is the only word such that prepending the reverse of its complement has no occurrence of the factor $10$.
On the other hand, $0\dots0$ must be mapped to $10\dots10$ if the length is even, and $010\dots10$ if it is odd.

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 $w$.