- FindStat

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Identifier

Mp00269: Binary words —flag zeros to zeros⟶ Binary words

Images

0 => 0

1 => 1

00 => 01

01 => 10

10 => 00

11 => 11

000 => 011

001 => 101

010 => 000

011 => 110

100 => 010

101 => 100

110 => 001

111 => 111

0000 => 0111

0001 => 1011

0010 => 0001

0011 => 1101

0100 => 0100

0101 => 1000

0110 => 0010

0111 => 1110

1000 => 0110

1001 => 1010

1010 => 0000

1011 => 1100

1100 => 0101

1101 => 1001

1110 => 0011

1111 => 1111

00000 => 01111

00001 => 10111

00010 => 00011

00011 => 11011

00100 => 01001

00101 => 10001

00110 => 00101

00111 => 11101

01000 => 01100

01001 => 10100

01010 => 00000

01011 => 11000

01100 => 01010

01101 => 10010

01110 => 00110

01111 => 11110

10000 => 01110

10001 => 10110

10010 => 00010

10011 => 11010

10100 => 01000

10101 => 10000

10110 => 00100

10111 => 11100

11000 => 01101

11001 => 10101

11010 => 00001

11011 => 11001

11100 => 01011

11101 => 10011

11110 => 00111

11111 => 11111

000000 => 011111

000001 => 101111

000010 => 000111

000011 => 110111

000100 => 010011

000101 => 100011

000110 => 001011

000111 => 111011

001000 => 011001

001001 => 101001

001010 => 000001

001011 => 110001

001100 => 010101

001101 => 100101

001110 => 001101

001111 => 111101

010000 => 011100

010001 => 101100

010010 => 000100

010011 => 110100

010100 => 010000

010101 => 100000

010110 => 001000

010111 => 111000

011000 => 011010

011001 => 101010

011010 => 000010

011011 => 110010

011100 => 010110

011101 => 100110

011110 => 001110

011111 => 111110

100000 => 011110

100001 => 101110

100010 => 000110

100011 => 110110

100100 => 010010

100101 => 100010

>>> Load all 2761 entries. <<<

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

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 $0\dots 0$.

Properties

bijective, graded

Sage code

def mapping(w):
    zeros = []
    letter = 1
    z = 0
    for e in reversed(w):
        if e == letter:
            z += 1
        else:
            zeros.append(z)
            z = 0
            letter = 1 - letter
    result = [1]*len(w)
    last = -1
    for z in zeros:
        last = z + last + 1
        result[last] = 0
    return Words([0,1])(result)

Weight

Created

Dec 02, 2022 at 00:03 by Martin Rubey

Updated

Dec 02, 2022 at 00:03 by Martin Rubey