- FindStat

Identifier

Mp00105: Binary words —complement⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words

Images

0 => 1 => 1 => 1

1 => 0 => 0 => 0

00 => 11 => 11 => 10

01 => 10 => 00 => 01

10 => 01 => 10 => 11

11 => 00 => 01 => 00

000 => 111 => 111 => 101

001 => 110 => 001 => 011

010 => 101 => 100 => 110

011 => 100 => 010 => 000

100 => 011 => 110 => 100

101 => 010 => 000 => 010

110 => 001 => 101 => 111

111 => 000 => 011 => 001

0000 => 1111 => 1111 => 1010

0001 => 1110 => 0011 => 0110

0010 => 1101 => 1001 => 1100

0011 => 1100 => 0101 => 0000

0100 => 1011 => 1100 => 1001

0101 => 1010 => 0000 => 0101

0110 => 1001 => 1010 => 1111

0111 => 1000 => 0110 => 0011

1000 => 0111 => 1110 => 1011

1001 => 0110 => 0010 => 0111

1010 => 0101 => 1000 => 1101

1011 => 0100 => 0100 => 0001

1100 => 0011 => 1101 => 1000

1101 => 0010 => 0001 => 0100

1110 => 0001 => 1011 => 1110

1111 => 0000 => 0111 => 0010

00000 => 11111 => 11111 => 10101

00001 => 11110 => 00111 => 01101

00010 => 11101 => 10011 => 11001

00011 => 11100 => 01011 => 00001

00100 => 11011 => 11001 => 10011

00101 => 11010 => 00001 => 01011

00110 => 11001 => 10101 => 11111

00111 => 11000 => 01101 => 00111

01000 => 10111 => 11100 => 10110

01001 => 10110 => 00100 => 01110

01010 => 10101 => 10000 => 11010

01011 => 10100 => 01000 => 00010

01100 => 10011 => 11010 => 10000

01101 => 10010 => 00010 => 01000

01110 => 10001 => 10110 => 11100

01111 => 10000 => 01110 => 00100

10000 => 01111 => 11110 => 10100

10001 => 01110 => 00110 => 01100

10010 => 01101 => 10010 => 11000

10011 => 01100 => 01010 => 00000

10100 => 01011 => 11000 => 10010

10101 => 01010 => 00000 => 01010

10110 => 01001 => 10100 => 11110

10111 => 01000 => 01100 => 00110

11000 => 00111 => 11101 => 10111

11001 => 00110 => 00101 => 01111

11010 => 00101 => 10001 => 11011

11011 => 00100 => 01001 => 00011

11100 => 00011 => 11011 => 10001

11101 => 00010 => 00011 => 01001

11110 => 00001 => 10111 => 11101

11111 => 00000 => 01111 => 00101

000000 => 111111 => 111111 => 101010

000001 => 111110 => 001111 => 011010

000010 => 111101 => 100111 => 110010

000011 => 111100 => 010111 => 000010

000100 => 111011 => 110011 => 100110

000101 => 111010 => 000011 => 010110

000110 => 111001 => 101011 => 111110

000111 => 111000 => 011011 => 001110

001000 => 110111 => 111001 => 101100

001001 => 110110 => 001001 => 011100

001010 => 110101 => 100001 => 110100

001011 => 110100 => 010001 => 000100

001100 => 110011 => 110101 => 100000

001101 => 110010 => 000101 => 010000

001110 => 110001 => 101101 => 111000

001111 => 110000 => 011101 => 001000

010000 => 101111 => 111100 => 101001

010001 => 101110 => 001100 => 011001

010010 => 101101 => 100100 => 110001

010011 => 101100 => 010100 => 000001

010100 => 101011 => 110000 => 100101

010101 => 101010 => 000000 => 010101

010110 => 101001 => 101000 => 111101

010111 => 101000 => 011000 => 001101

011000 => 100111 => 111010 => 101111

011001 => 100110 => 001010 => 011111

011010 => 100101 => 100010 => 110111

011011 => 100100 => 010010 => 000111

011100 => 100011 => 110110 => 100011

011101 => 100010 => 000110 => 010011

011110 => 100001 => 101110 => 111011

011111 => 100000 => 011110 => 001011

100000 => 011111 => 111110 => 101011

100001 => 011110 => 001110 => 011011

100010 => 011101 => 100110 => 110011

100011 => 011100 => 010110 => 000011

100100 => 011011 => 110010 => 100111

100101 => 011010 => 000010 => 010111

100110 => 011001 => 101010 => 111111

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Map

complement

Description

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

Map

flag zeros to zeros

Description

Return a binary word of the same length, such that the number of zeros equals the number of occurrences of $10$ in the word obtained from the original word by prepending the reverse of the complement.
For example, the image of the word $w=1\dots 1$ is $1\dots 1$, because $0\dots 01\dots 1$ has no occurrences of $10$. The words $10\dots 10$ and $010\dots 10$ have image $0\dots 0$.

Map

alternating inverse

Description

Sends a binary word $w_1\cdots w_m$ to the binary word $v_1 \cdots v_m$ with $v_i = w_i$ if $i$ is odd and $v_i = 1 - w_i$ if $i$ is even.
This map is used in [1], see Definitions 3.2 and 5.1.