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Identifier
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
Images
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111010001001=>110110100111 111010100001=>110101100111 111100000101=>110010101111 111100001001=>110110101111 111100010001=>110100101111 111100100001=>110101101111 111101000001=>110101001111 10101010101010=>01100110011001 10101010101100=>01000110011001 10101010110100=>01001110011001 10101010111000=>01011110011001 10101011010100=>01001100011001 10101011100100=>01001000011001 10101011101000=>01011000011001 10101011110000=>01010000011001 10101101010100=>01001100111001 10101110010100=>01001101111001 10101110100100=>01001001111001 10101110101000=>01011001111001 10101110110000=>01010001111001 10101111000100=>01001011111001 10101111010000=>01010011111001 10110101010100=>01001100110001 10110111100000=>01010111110001 10111001010100=>01001100100001 10111010010100=>01001101100001 10111010100100=>01001001100001 10111011000100=>01001011100001 10111011010000=>01010011100001 10111011100000=>01010111100001 10111100010100=>01001101000001 10111100110000=>01010001000001 10111101000100=>01001011000001 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11100111010000=>01010011110111 11100111100000=>01010111110111 11101001010100=>01001100100111 11101001011000=>01011100100111 11101001100100=>01001000100111 11101001101000=>01011000100111 11101001110000=>01010000100111 11101010010100=>01001101100111 11101010011000=>01011101100111 11101010100100=>01001001100111 11101010101000=>01011001100111 11101010110000=>01010001100111 11101011000100=>01001011100111 11101011001000=>01011011100111 11101011010000=>01010011100111 11101011100000=>01010111100111 11101100010100=>01001101000111 11101100011000=>01011101000111 11101100100100=>01001001000111 11101100101000=>01011001000111 11101100110000=>01010001000111 11101101000010=>01101011000111 11101101000100=>01001011000111 11101101001000=>01011011000111 11101101010000=>01010011000111 11101101100000=>01010111000111 11101110000010=>01101010000111 11101110000100=>01001010000111 11101110001000=>01011010000111 11101110010000=>01010010000111 11101110100000=>01010110000111 11101111000000=>01010100000111 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1111101100000010=>1001010100011111 1111110001000010=>1001010010111111 1111110010000010=>1001010110111111 1111110100000010=>1001010100111111 1111111000000010=>1001010101111111 101010101010101010=>011001100110011001 101010101010101100=>010001100110011001 101010101010110100=>010011100110011001 101010101011010100=>010011000110011001 101010101101010100=>010011001110011001 101010110101010100=>010011001100011001 101011010101010100=>010011001100111001 101101010101010100=>010011001100110001 110101010101010100=>010011001100110011 101010101010111000=>010111100110011001 101010101011100100=>010010000110011001 101010101110010100=>010011011110011001 101010111001010100=>010011001000011001 101011100101010100=>010011001101111001 101110010101010100=>010011001100100001 111001010101010100=>010011001100110111 111111110000000010=>011010101011111111 101111111100000000=>010101010000000001 111111101000000010=>011010101001111111 101111111010000000=>010101011000000001 111111100100000010=>011010101101111111 111111011000000010=>011010101000111111 101111111001000000=>010101001000000001 101111110110000000=>010101011100000001 10101010101010101010=>10011001100110011001 10101010101010101100=>10111001100110011001 10101010101010110100=>10110001100110011001 10101010101011010100=>10110011100110011001 10101010101101010100=>10110011000110011001 10101010110101010100=>10110011001110011001 10101011010101010100=>10110011001100011001 10101101010101010100=>10110011001100111001 10110101010101010100=>10110011001100110001 11010101010101010100=>10110011001100110011 11111111100000000010=>10010101010111111111 10111111111000000000=>10101010100000000001 11111111010000000010=>10010101010011111111 10111111110100000000=>10101010110000000001 1111111111000000000010=>0110101010101111111111 1011111111110000000000=>0101010101000000000001
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 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.
Properties
bijective, graded
Sage code
def mapping(w):
    n = len(w)
    zeros = [-1] + [i for i in range(n) if w[i] == 0]
    zeros = [zeros[i] - zeros[i-1] - 1 for i in range(1, len(zeros))]
    result = []
    letter = 1
    for b in zeros:
        result += [letter]*b
        if letter == 0:
            result += [0, 1]
        letter = 1 - letter
    result += [letter]*(n-len(result))
    return Words([0,1])(reversed(result))
Weight
40
Created
Dec 01, 2022 at 22:36 by Martin Rubey
Updated
Dec 01, 2022 at 22:36 by Martin Rubey