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Identifier
Values
=>
0=>2 1=>2 00=>1 01=>2 10=>2 11=>1 000=>1 001=>4 010=>1 011=>4 100=>4 101=>1 110=>4 111=>1 0000=>1 0001=>6 0010=>3 0011=>6 0100=>3 0101=>1 0110=>3 0111=>6 1000=>6 1001=>3 1010=>1 1011=>3 1100=>6 1101=>3 1110=>6 1111=>1 00000=>1 00001=>12 00010=>5 00011=>12 00100=>1 00101=>12 00110=>5 00111=>12 01000=>5 01001=>2 01010=>1 01011=>12 01100=>5 01101=>2 01110=>5 01111=>12 10000=>12 10001=>5 10010=>2 10011=>5 10100=>12 10101=>1 10110=>2 10111=>5 11000=>12 11001=>5 11010=>12 11011=>1 11100=>12 11101=>5 11110=>12 11111=>1 000000=>1 000001=>20 000010=>11 000011=>20 000100=>3 000101=>20 000110=>11 000111=>20 001000=>3 001001=>1 001010=>11 001011=>20 001100=>3 001101=>20 001110=>11 001111=>20 010000=>11 010001=>3 010010=>1 010011=>20 010100=>11 010101=>1 010110=>11 010111=>20 011000=>11 011001=>3 011010=>11 011011=>1 011100=>11 011101=>3 011110=>11 011111=>20 100000=>20 100001=>11 100010=>3 100011=>11 100100=>1 100101=>11 100110=>3 100111=>11 101000=>20 101001=>11 101010=>1 101011=>11 101100=>20 101101=>1 101110=>3 101111=>11 110000=>20 110001=>11 110010=>20 110011=>3 110100=>20 110101=>11 110110=>1 110111=>3 111000=>20 111001=>11 111010=>20 111011=>3 111100=>20 111101=>11 111110=>20 111111=>1 0000000=>1 0000001=>40 0000010=>19 0000011=>40 0000100=>6 0000101=>40 0000110=>19 0000111=>40 0001000=>1 0001001=>40 0001010=>19 0001011=>40 0001100=>6 0001101=>40 0001110=>19 0001111=>40 0010000=>6 0010001=>2 0010010=>1 0010011=>40 0010100=>6 0010101=>40 0010110=>19 0010111=>40 0011000=>6 0011001=>2 0011010=>19 0011011=>40 0011100=>6 0011101=>40 0011110=>19 0011111=>40 0100000=>19 0100001=>8 0100010=>1 0100011=>40 0100100=>1 0100101=>8 0100110=>19 0100111=>40 0101000=>19 0101001=>8 0101010=>1 0101011=>40 0101100=>19 0101101=>8 0101110=>19 0101111=>40 0110000=>19 0110001=>8 0110010=>19 0110011=>2 0110100=>19 0110101=>8 0110110=>1 0110111=>40 0111000=>19 0111001=>8 0111010=>19 0111011=>2 0111100=>19 0111101=>8 0111110=>19 0111111=>40 1000000=>40 1000001=>19 1000010=>8 1000011=>19 1000100=>2 1000101=>19 1000110=>8 1000111=>19 1001000=>40 1001001=>1 1001010=>8 1001011=>19 1001100=>2 1001101=>19 1001110=>8 1001111=>19 1010000=>40 1010001=>19 1010010=>8 1010011=>19 1010100=>40 1010101=>1 1010110=>8 1010111=>19 1011000=>40 1011001=>19 1011010=>8 1011011=>1 1011100=>40 1011101=>1 1011110=>8 1011111=>19 1100000=>40 1100001=>19 1100010=>40 1100011=>6 1100100=>40 1100101=>19 1100110=>2 1100111=>6 1101000=>40 1101001=>19 1101010=>40 1101011=>6 1101100=>40 1101101=>1 1101110=>2 1101111=>6 1110000=>40 1110001=>19 1110010=>40 1110011=>6 1110100=>40 1110101=>19 1110110=>40 1110111=>1 1111000=>40 1111001=>19 1111010=>40 1111011=>6 1111100=>40 1111101=>19 1111110=>40 1111111=>1 00000000=>1 00000001=>74 00000010=>41 00000011=>74 00000100=>11 00000101=>74 00000110=>41 00000111=>74 00001000=>3 00001001=>74 00001010=>41 00001011=>74 00001100=>11 00001101=>74 00001110=>41 00001111=>74 00010000=>3 00010001=>1 00010010=>41 00010011=>74 00010100=>11 00010101=>74 00010110=>41 00010111=>74 00011000=>3 00011001=>74 00011010=>41 00011011=>74 00011100=>11 00011101=>74 00011110=>41 00011111=>74 00100000=>11 00100001=>4 00100010=>1 00100011=>74 00100100=>1 00100101=>74 00100110=>41 00100111=>74 00101000=>11 00101001=>4 00101010=>41 00101011=>74 00101100=>11 00101101=>74 00101110=>41 00101111=>74 00110000=>11 00110001=>4 00110010=>41 00110011=>1 00110100=>11 00110101=>74 00110110=>41 00110111=>74 00111000=>11 00111001=>4 00111010=>41 00111011=>74 00111100=>11 00111101=>74 00111110=>41 00111111=>74 01000000=>41 01000001=>13 01000010=>4 01000011=>74 01000100=>1 01000101=>13 01000110=>41 01000111=>74 01001000=>41 01001001=>1 01001010=>4 01001011=>74 01001100=>41 01001101=>13 01001110=>41 01001111=>74 01010000=>41 01010001=>13 01010010=>4 01010011=>74 01010100=>41 01010101=>1 01010110=>41 01010111=>74 01011000=>41 01011001=>13 01011010=>4 01011011=>74 01011100=>41 01011101=>13 01011110=>41 01011111=>74 01100000=>41 01100001=>13 01100010=>41 01100011=>4 01100100=>41 01100101=>13 01100110=>1 01100111=>74 01101000=>41 01101001=>13 01101010=>41 01101011=>4 01101100=>41 01101101=>1 01101110=>41 01101111=>74 01110000=>41 01110001=>13 01110010=>41 01110011=>4 01110100=>41 01110101=>13 01110110=>41 01110111=>1 01111000=>41 01111001=>13 01111010=>41 01111011=>4 01111100=>41 01111101=>13 01111110=>41 01111111=>74 10000000=>74 10000001=>41 10000010=>13 10000011=>41 10000100=>4 10000101=>41 10000110=>13 10000111=>41 10001000=>1 10001001=>41 10001010=>13 10001011=>41 10001100=>4 10001101=>41 10001110=>13 10001111=>41 10010000=>74 10010001=>41 10010010=>1 10010011=>41 10010100=>4 10010101=>41 10010110=>13 10010111=>41 10011000=>74 10011001=>1 10011010=>13 10011011=>41 10011100=>4 10011101=>41 10011110=>13 10011111=>41 10100000=>74 10100001=>41 10100010=>13 10100011=>41 10100100=>74 10100101=>4 10100110=>13 10100111=>41 10101000=>74 10101001=>41 10101010=>1 10101011=>41 10101100=>74 10101101=>4 10101110=>13 10101111=>41 10110000=>74 10110001=>41 10110010=>13 10110011=>41 10110100=>74 10110101=>4 10110110=>1 10110111=>41 10111000=>74 10111001=>41 10111010=>13 10111011=>1 10111100=>74 10111101=>4 10111110=>13 10111111=>41 11000000=>74 11000001=>41 11000010=>74 11000011=>11 11000100=>74 11000101=>41 11000110=>4 11000111=>11 11001000=>74 11001001=>41 11001010=>74 11001011=>11 11001100=>1 11001101=>41 11001110=>4 11001111=>11 11010000=>74 11010001=>41 11010010=>74 11010011=>11 11010100=>74 11010101=>41 11010110=>4 11010111=>11 11011000=>74 11011001=>41 11011010=>74 11011011=>1 11011100=>74 11011101=>1 11011110=>4 11011111=>11 11100000=>74 11100001=>41 11100010=>74 11100011=>11 11100100=>74 11100101=>41 11100110=>74 11100111=>3 11101000=>74 11101001=>41 11101010=>74 11101011=>11 11101100=>74 11101101=>41 11101110=>1 11101111=>3 11110000=>74 11110001=>41 11110010=>74 11110011=>11 11110100=>74 11110101=>41 11110110=>74 11110111=>3 11111000=>74 11111001=>41 11111010=>74 11111011=>11 11111100=>74 11111101=>41 11111110=>74 11111111=>1 000000000=>1 000000001=>148 000000010=>77 000000011=>148 000000100=>23 000000101=>148 000000110=>77 000000111=>148 000001000=>6 000001001=>148 000001010=>77 000001011=>148 000001100=>23 000001101=>148 000001110=>77 000001111=>148 000010000=>1 000010001=>148 000010010=>77 000010011=>148 000010100=>23 000010101=>148 000010110=>77 000010111=>148 000011000=>6 000011001=>148 000011010=>77 000011011=>148 000011100=>23 000011101=>148 000011110=>77 000011111=>148 000100000=>6 000100001=>2 000100010=>1 000100011=>148 000100100=>23 000100101=>148 000100110=>77 000100111=>148 000101000=>6 000101001=>148 000101010=>77 000101011=>148 000101100=>23 000101101=>148 000101110=>77 000101111=>148 000110000=>6 000110001=>2 000110010=>77 000110011=>148 000110100=>23 000110101=>148 000110110=>77 000110111=>148 000111000=>6 000111001=>148 000111010=>77 000111011=>148 000111100=>23 000111101=>148 000111110=>77 000111111=>148 001000000=>23 001000001=>7 001000010=>2 001000011=>148 001000100=>1 001000101=>148 001000110=>77 001000111=>148 001001000=>23 001001001=>1 001001010=>77 001001011=>148 001001100=>23 001001101=>148 001001110=>77 001001111=>148 001010000=>23 001010001=>7 001010010=>2 001010011=>148 001010100=>23 001010101=>148 001010110=>77 001010111=>148 001011000=>23 001011001=>7 001011010=>77 001011011=>148 001011100=>23 001011101=>148 001011110=>77 001011111=>148 001100000=>23 001100001=>7 001100010=>77 001100011=>2 001100100=>23 001100101=>148 001100110=>1 001100111=>148 001101000=>23 001101001=>7 001101010=>77 001101011=>148 001101100=>23 001101101=>148 001101110=>77 001101111=>148 001110000=>23 001110001=>7 001110010=>77 001110011=>2 001110100=>23 001110101=>148 001110110=>77 001110111=>148 001111000=>23 001111001=>7 001111010=>77 001111011=>148 001111100=>23 001111101=>148 001111110=>77 001111111=>148 010000000=>77 010000001=>30 010000010=>6 010000011=>148 010000100=>2 010000101=>30 010000110=>77 010000111=>148 010001000=>1 010001001=>30 010001010=>6 010001011=>148 010001100=>77 010001101=>30 010001110=>77 010001111=>148 010010000=>77 010010001=>30 010010010=>1 010010011=>148 010010100=>2 010010101=>30 010010110=>77 010010111=>148 010011000=>77 010011001=>30 010011010=>6 010011011=>148 010011100=>77 010011101=>30 010011110=>77 010011111=>148 010100000=>77 010100001=>30 010100010=>6 010100011=>148 010100100=>77 010100101=>2 010100110=>77 010100111=>148 010101000=>77 010101001=>30 010101010=>1 010101011=>148 010101100=>77 010101101=>30 010101110=>77 010101111=>148 010110000=>77 010110001=>30 010110010=>6 010110011=>148 010110100=>77 010110101=>2 010110110=>77 010110111=>148 010111000=>77 010111001=>30 010111010=>6 010111011=>148 010111100=>77 010111101=>30 010111110=>77 010111111=>148 011000000=>77 011000001=>30 011000010=>77 011000011=>7 011000100=>77 011000101=>30 011000110=>2 011000111=>148 011001000=>77 011001001=>30 011001010=>77 011001011=>7 011001100=>1 011001101=>30 011001110=>77 011001111=>148 011010000=>77 011010001=>30 011010010=>77 011010011=>7 011010100=>77 011010101=>30 011010110=>2 011010111=>148 011011000=>77 011011001=>30 011011010=>77 011011011=>1 011011100=>77 011011101=>30 011011110=>77 011011111=>148 011100000=>77 011100001=>30 011100010=>77 011100011=>7 011100100=>77 011100101=>30 011100110=>77 011100111=>2 011101000=>77 011101001=>30 011101010=>77 011101011=>7 011101100=>77 011101101=>30 011101110=>1 011101111=>148 011110000=>77 011110001=>30 011110010=>77 011110011=>7 011110100=>77 011110101=>30 011110110=>77 011110111=>2 011111000=>77 011111001=>30 011111010=>77 011111011=>7 011111100=>77 011111101=>30 011111110=>77 011111111=>148 100000000=>148 100000001=>77 100000010=>30 100000011=>77 100000100=>7 100000101=>77 100000110=>30 100000111=>77 100001000=>2 100001001=>77 100001010=>30 100001011=>77 100001100=>7 100001101=>77 100001110=>30 100001111=>77 100010000=>148 100010001=>1 100010010=>30 100010011=>77 100010100=>7 100010101=>77 100010110=>30 100010111=>77 100011000=>2 100011001=>77 100011010=>30 100011011=>77 100011100=>7 100011101=>77 100011110=>30 100011111=>77 100100000=>148 100100001=>77 100100010=>30 100100011=>77 100100100=>1 100100101=>77 100100110=>30 100100111=>77 100101000=>148 100101001=>2 100101010=>30 100101011=>77 100101100=>7 100101101=>77 100101110=>30 100101111=>77 100110000=>148 100110001=>77 100110010=>30 100110011=>1 100110100=>7 100110101=>77 100110110=>30 100110111=>77 100111000=>148 100111001=>2 100111010=>30 100111011=>77 100111100=>7 100111101=>77 100111110=>30 100111111=>77 101000000=>148 101000001=>77 101000010=>30 101000011=>77 101000100=>148 101000101=>6 101000110=>30 101000111=>77 101001000=>148 101001001=>77 101001010=>2 101001011=>77 101001100=>148 101001101=>6 101001110=>30 101001111=>77 101010000=>148 101010001=>77 101010010=>30 101010011=>77 101010100=>148 101010101=>1 101010110=>30 101010111=>77 101011000=>148 101011001=>77 101011010=>2 101011011=>77 101011100=>148 101011101=>6 101011110=>30 101011111=>77 101100000=>148 101100001=>77 101100010=>30 101100011=>77 101100100=>148 101100101=>6 101100110=>30 101100111=>77 101101000=>148 101101001=>77 101101010=>30 101101011=>2 101101100=>148 101101101=>1 101101110=>30 101101111=>77 101110000=>148 101110001=>77 101110010=>30 101110011=>77 101110100=>148 101110101=>6 101110110=>30 101110111=>1 101111000=>148 101111001=>77 101111010=>30 101111011=>2 101111100=>148 101111101=>6 101111110=>30 101111111=>77 110000000=>148 110000001=>77 110000010=>148 110000011=>23 110000100=>148 110000101=>77 110000110=>7 110000111=>23 110001000=>148 110001001=>77 110001010=>148 110001011=>23 110001100=>2 110001101=>77 110001110=>7 110001111=>23 110010000=>148 110010001=>77 110010010=>148 110010011=>23 110010100=>148 110010101=>77 110010110=>7 110010111=>23 110011000=>148 110011001=>1 110011010=>148 110011011=>23 110011100=>2 110011101=>77 110011110=>7 110011111=>23 110100000=>148 110100001=>77 110100010=>148 110100011=>23 110100100=>148 110100101=>77 110100110=>7 110100111=>23 110101000=>148 110101001=>77 110101010=>148 110101011=>23 110101100=>148 110101101=>2 110101110=>7 110101111=>23 110110000=>148 110110001=>77 110110010=>148 110110011=>23 110110100=>148 110110101=>77 110110110=>1 110110111=>23 110111000=>148 110111001=>77 110111010=>148 110111011=>1 110111100=>148 110111101=>2 110111110=>7 110111111=>23 111000000=>148 111000001=>77 111000010=>148 111000011=>23 111000100=>148 111000101=>77 111000110=>148 111000111=>6 111001000=>148 111001001=>77 111001010=>148 111001011=>23 111001100=>148 111001101=>77 111001110=>2 111001111=>6 111010000=>148 111010001=>77 111010010=>148 111010011=>23 111010100=>148 111010101=>77 111010110=>148 111010111=>6 111011000=>148 111011001=>77 111011010=>148 111011011=>23 111011100=>148 111011101=>1 111011110=>2 111011111=>6 111100000=>148 111100001=>77 111100010=>148 111100011=>23 111100100=>148 111100101=>77 111100110=>148 111100111=>6 111101000=>148 111101001=>77 111101010=>148 111101011=>23 111101100=>148 111101101=>77 111101110=>148 111101111=>1 111110000=>148 111110001=>77 111110010=>148 111110011=>23 111110100=>148 111110101=>77 111110110=>148 111110111=>6 111111000=>148 111111001=>77 111111010=>148 111111011=>23 111111100=>148 111111101=>77 111111110=>148 111111111=>1
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Description
The number of binary words with the same proper border set.
The proper border set of a binary word $w$ is the set of proper prefixes which are also suffixes of $w$.
For example, $0010000010$, $0010100010$ and $0010110010$ are the words with proper border set $\{0, 0010\}$, whereas $0010010010$ has proper border set $\{0, 0010, 0010010\}$.
Code
@cached_function
def proper_border_set(w):
    B = []
    for i in range(1, len(w)):
        if w[:i] == w[-i:]:
            B.append(w[:i])
    return B

def statistic(w):
    B = proper_border_set(w)
    return sum(1 for u in Words([0,1], len(w)) if B == proper_border_set(u))

Created
Dec 21, 2022 at 11:43 by Martin Rubey
Updated
Dec 21, 2022 at 11:43 by Martin Rubey