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Identifier
Mp00269: Binary words flag zeros to zerosBinary words
Images
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001111100010=>000110111101 100111100100=>010010111010 001111100100=>010010111101 010011101010=>000000110100 001011101010=>000000110001 100111101000=>011000111010 001111101000=>011000111101 000111101010=>000000111011 100011101100=>010100110110 010011101100=>010100110100 001011101100=>010100110001 000111101100=>010100111011 000011101110=>001100110111 101001110100=>010001101000 001001110110=>001001101001 001101110010=>000101100101 100101110100=>010001100010 001101110100=>010001100101 000101110110=>001001100011 001011110010=>000101110001 100111110000=>011101111010 001111110000=>011101111101 100011110100=>010001110110 010011110100=>010001110100 001011110100=>010001110001 000111110100=>010001111011 000011110110=>001001110111 110001111000=>011011101101 101001111000=>011011101000 001001111010=>000011101001 100101111000=>011011100010 001101111000=>011011100101 100011111000=>011011110110 001011111000=>011011110001 000011111010=>000011110111 010001111100=>010111101100 001001111100=>010111101001 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110011000101=>100011010101 110011001001=>101001010101 110011010001=>101100010101 110011100001=>101110110101 110100010101=>100000110001 110100011001=>101010110001 110100100101=>100010010001 110100101001=>101000010001 110100110001=>101101010001 110101000101=>100011000001 110101001001=>101001000001 110101010001=>101100000001 110101100001=>101110100001 110110000101=>100011101001 110110001001=>101001101001 110110010001=>101100101001 110110100001=>101110001001 110111000001=>101111011001 111000010101=>100000111011 111000101001=>101000011011 111001000101=>100011001011 111001001001=>101001001011 111001010001=>101100001011 111010001001=>101001100011 111010100001=>101110000011 111100000101=>100011110111 111100001001=>101001110111 111100010001=>101100110111 111100100001=>101110010111 111101000001=>101111000111 10101010101010=>00000000000000 10101010101100=>01010000000000 10101010110100=>01000100000000 10101010111000=>01101100000000 10101011010100=>01000001000000 10101011100100=>01001011000000 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1011010101010100=>0100000000000100 1011100101010100=>0100000000101100 1011101001010100=>0100000010001100 1011110001010100=>0100000011011100 1011110111000000=>0111110110011100 1011111010100000=>0111100000111100 1011111011000000=>0111110100111100 1011111100010000=>0111001101111100 1011111100100000=>0111100101111100 1011111101000000=>0111110001111100 1011111110000000=>0111111011111100 1101010101010100=>0100000000000001 1110010101010100=>0100000000001011 1110100101010100=>0100000000100011 1111000101010100=>0100000000110111 1111011100000010=>0001111101100111 1111101010000010=>0001111000001111 1111101100000010=>0001111101001111 1111110001000010=>0001110011011111 1111110010000010=>0001111001011111 1111110100000010=>0001111100011111 1111111000000010=>0001111110111111 101010101010101010=>000000000000000000 101010101010101100=>010100000000000000 101010101010110100=>010001000000000000 101010101011010100=>010000010000000000 101010101101010100=>010000000100000000 101010110101010100=>010000000001000000 101011010101010100=>010000000000010000 101101010101010100=>010000000000000100 110101010101010100=>010000000000000001 101010101010111000=>011011000000000000 101010101011100100=>010010110000000000 101010101110010100=>010000101100000000 101010111001010100=>010000001011000000 101011100101010100=>010000000010110000 101110010101010100=>010000000000101100 111001010101010100=>010000000000001011 111111110000000010=>000111111101111111 101111111100000000=>011111110111111100 111111101000000010=>000111111000111111 101111111010000000=>011111100011111100 111111100100000010=>000111110010111111 111111011000000010=>000111111010011111 101111111001000000=>011111001011111100 101111110110000000=>011111101001111100 10101010101010101010=>00000000000000000000 10101010101010101100=>01010000000000000000 10101010101010110100=>01000100000000000000 10101010101011010100=>01000001000000000000 10101010101101010100=>01000000010000000000 10101010110101010100=>01000000000100000000 10101011010101010100=>01000000000001000000 10101101010101010100=>01000000000000010000 10110101010101010100=>01000000000000000100 11010101010101010100=>01000000000000000001 11111111100000000010=>00011111111011111111 10111111111000000000=>01111111101111111100 11111111010000000010=>00011111110001111111 10111111110100000000=>01111111000111111100 1111111111000000000010=>0001111111110111111111 1011111111110000000000=>0111111111011111111100
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
40
Created
Dec 02, 2022 at 00:03 by Martin Rubey
Updated
Dec 02, 2022 at 00:03 by Martin Rubey