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Identifier

Mp00273: Binary words —Gray previous⟶ Binary words

Images

0 => 1

1 => 0

00 => 01

01 => 11

10 => 00

11 => 10

000 => 001

001 => 101

010 => 110

011 => 010

100 => 000

101 => 111

110 => 100

111 => 011

0000 => 0001

0001 => 1001

0010 => 1010

0011 => 0010

0100 => 1100

0101 => 0111

0110 => 0100

0111 => 1111

1000 => 0000

1001 => 1101

1010 => 1110

1011 => 0011

1100 => 1000

1101 => 0101

1110 => 0110

1111 => 1011

00000 => 00001

00001 => 10001

00010 => 10010

00011 => 00010

00100 => 10100

00101 => 00111

00110 => 00100

00111 => 10111

01000 => 11000

01001 => 01101

01010 => 01110

01011 => 11011

01100 => 01000

01101 => 11101

01110 => 11110

01111 => 01011

10000 => 00000

10001 => 11001

10010 => 11010

10011 => 00011

10100 => 11100

10101 => 00101

10110 => 00110

10111 => 11111

11000 => 10000

11001 => 01001

11010 => 01010

11011 => 10011

11100 => 01100

11101 => 10101

11110 => 10110

11111 => 01111

000000 => 000001

000001 => 100001

000010 => 100010

000011 => 000010

000100 => 100100

000101 => 000111

000110 => 000100

000111 => 100111

001000 => 101000

001001 => 001101

001010 => 001110

001011 => 101011

001100 => 001000

001101 => 101101

001110 => 101110

001111 => 001011

010000 => 110000

010001 => 011001

010010 => 011010

010011 => 110011

010100 => 011100

010101 => 110101

010110 => 110110

010111 => 011111

011000 => 010000

011001 => 111001

011010 => 111010

011011 => 010011

011100 => 111100

011101 => 010101

011110 => 010110

011111 => 111111

100000 => 000000

100001 => 110001

100010 => 110010

100011 => 000011

100100 => 110100

100101 => 000101

100110 => 000110

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click to show experimental identities (only identities of compositions of up to three maps are shown)

Description

The previous element in the Gray code for binary words.
Let $w$ be a binary word. If the number of ones in $w$ is odd, return the binary word obtained by flipping the first bit of $w$. If $w$ has only zeros, return the word with the final bit flipped. Otherwise, return the word obtained by flipping the bit after the first one.

References

[1] wikipedia:Gray code

Properties

bijective

Sage code

def mapping(w):
    if not w:
        return w
    w = list(w)
    if is_odd(w.count(1)):
        w[0] = 1 - w[0]
        return Word(w)
    try:
        i = w.index(1)
        w[i + 1] = 1 - w[i + 1]
        return Word(w)
    except ValueError:
        w[-1] = 1
        return Word(w)

Weight

Created

Dec 08, 2022 at 17:42 by Martin Rubey

Updated

Dec 08, 2022 at 17:42 by Martin Rubey