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Identifier
Values
=>
0=>0 1=>0 00=>0 01=>0 10=>0 11=>0 000=>1 001=>0 010=>1 011=>0 100=>0 101=>1 110=>0 111=>1 0000=>1 0001=>1 0010=>0 0011=>0 0100=>1 0101=>1 0110=>0 0111=>0 1000=>0 1001=>0 1010=>1 1011=>1 1100=>0 1101=>0 1110=>1 1111=>1 00000=>2 00001=>1 00010=>1 00011=>1 00100=>2 00101=>0 00110=>0 00111=>0 01000=>1 01001=>1 01010=>2 01011=>1 01100=>0 01101=>0 01110=>2 01111=>0 10000=>0 10001=>2 10010=>0 10011=>0 10100=>1 10101=>2 10110=>1 10111=>1 11000=>0 11001=>0 11010=>0 11011=>2 11100=>1 11101=>1 11110=>1 11111=>2 000000=>2 000001=>2 000010=>1 000011=>1 000100=>1 000101=>1 000110=>1 000111=>1 001000=>2 001001=>2 001010=>0 001011=>0 001100=>0 001101=>0 001110=>0 001111=>0 010000=>1 010001=>1 010010=>1 010011=>1 010100=>2 010101=>2 010110=>1 010111=>1 011000=>0 011001=>0 011010=>0 011011=>0 011100=>2 011101=>2 011110=>0 011111=>0 100000=>0 100001=>0 100010=>2 100011=>2 100100=>0 100101=>0 100110=>0 100111=>0 101000=>1 101001=>1 101010=>2 101011=>2 101100=>1 101101=>1 101110=>1 101111=>1 110000=>0 110001=>0 110010=>0 110011=>0 110100=>0 110101=>0 110110=>2 110111=>2 111000=>1 111001=>1 111010=>1 111011=>1 111100=>1 111101=>1 111110=>2 111111=>2 0000000=>3 0000001=>2 0000010=>2 0000011=>2 0000100=>1 0000101=>1 0000110=>1 0000111=>1 0001000=>3 0001001=>1 0001010=>1 0001011=>1 0001100=>1 0001101=>1 0001110=>1 0001111=>1 0010000=>2 0010001=>2 0010010=>2 0010011=>2 0010100=>3 0010101=>0 0010110=>0 0010111=>0 0011000=>0 0011001=>0 0011010=>0 0011011=>0 0011100=>3 0011101=>0 0011110=>0 0011111=>0 0100000=>1 0100001=>1 0100010=>3 0100011=>1 0100100=>1 0100101=>1 0100110=>1 0100111=>1 0101000=>2 0101001=>2 0101010=>3 0101011=>2 0101100=>1 0101101=>1 0101110=>1 0101111=>1 0110000=>0 0110001=>0 0110010=>0 0110011=>0 0110100=>0 0110101=>0 0110110=>3 0110111=>0 0111000=>2 0111001=>2 0111010=>2 0111011=>2 0111100=>0 0111101=>0 0111110=>3 0111111=>0 1000000=>0 1000001=>3 1000010=>0 1000011=>0 1000100=>2 1000101=>2 1000110=>2 1000111=>2 1001000=>0 1001001=>3 1001010=>0 1001011=>0 1001100=>0 1001101=>0 1001110=>0 1001111=>0 1010000=>1 1010001=>1 1010010=>1 1010011=>1 1010100=>2 1010101=>3 1010110=>2 1010111=>2 1011000=>1 1011001=>1 1011010=>1 1011011=>1 1011100=>1 1011101=>3 1011110=>1 1011111=>1 1100000=>0 1100001=>0 1100010=>0 1100011=>3 1100100=>0 1100101=>0 1100110=>0 1100111=>0 1101000=>0 1101001=>0 1101010=>0 1101011=>3 1101100=>2 1101101=>2 1101110=>2 1101111=>2 1110000=>1 1110001=>1 1110010=>1 1110011=>1 1110100=>1 1110101=>1 1110110=>1 1110111=>3 1111000=>1 1111001=>1 1111010=>1 1111011=>1 1111100=>2 1111101=>2 1111110=>2 1111111=>3 00000000=>3 00000001=>3 00000010=>2 00000011=>2 00000100=>2 00000101=>2 00000110=>2 00000111=>2 00001000=>1 00001001=>1 00001010=>1 00001011=>1 00001100=>1 00001101=>1 00001110=>1 00001111=>1 00010000=>3 00010001=>3 00010010=>1 00010011=>1 00010100=>1 00010101=>1 00010110=>1 00010111=>1 00011000=>1 00011001=>1 00011010=>1 00011011=>1 00011100=>1 00011101=>1 00011110=>1 00011111=>1 00100000=>2 00100001=>2 00100010=>2 00100011=>2 00100100=>2 00100101=>2 00100110=>2 00100111=>2 00101000=>3 00101001=>3 00101010=>0 00101011=>0 00101100=>0 00101101=>0 00101110=>0 00101111=>0 00110000=>0 00110001=>0 00110010=>0 00110011=>0 00110100=>0 00110101=>0 00110110=>0 00110111=>0 00111000=>3 00111001=>3 00111010=>0 00111011=>0 00111100=>0 00111101=>0 00111110=>0 00111111=>0 01000000=>1 01000001=>1 01000010=>1 01000011=>1 01000100=>3 01000101=>3 01000110=>1 01000111=>1 01001000=>1 01001001=>1 01001010=>1 01001011=>1 01001100=>1 01001101=>1 01001110=>1 01001111=>1 01010000=>2 01010001=>2 01010010=>2 01010011=>2 01010100=>3 01010101=>3 01010110=>2 01010111=>2 01011000=>1 01011001=>1 01011010=>1 01011011=>1 01011100=>1 01011101=>1 01011110=>1 01011111=>1 01100000=>0 01100001=>0 01100010=>0 01100011=>0 01100100=>0 01100101=>0 01100110=>0 01100111=>0 01101000=>0 01101001=>0 01101010=>0 01101011=>0 01101100=>3 01101101=>3 01101110=>0 01101111=>0 01110000=>2 01110001=>2 01110010=>2 01110011=>2 01110100=>2 01110101=>2 01110110=>2 01110111=>2 01111000=>0 01111001=>0 01111010=>0 01111011=>0 01111100=>3 01111101=>3 01111110=>0 01111111=>0 10000000=>0 10000001=>0 10000010=>3 10000011=>3 10000100=>0 10000101=>0 10000110=>0 10000111=>0 10001000=>2 10001001=>2 10001010=>2 10001011=>2 10001100=>2 10001101=>2 10001110=>2 10001111=>2 10010000=>0 10010001=>0 10010010=>3 10010011=>3 10010100=>0 10010101=>0 10010110=>0 10010111=>0 10011000=>0 10011001=>0 10011010=>0 10011011=>0 10011100=>0 10011101=>0 10011110=>0 10011111=>0 10100000=>1 10100001=>1 10100010=>1 10100011=>1 10100100=>1 10100101=>1 10100110=>1 10100111=>1 10101000=>2 10101001=>2 10101010=>3 10101011=>3 10101100=>2 10101101=>2 10101110=>2 10101111=>2 10110000=>1 10110001=>1 10110010=>1 10110011=>1 10110100=>1 10110101=>1 10110110=>1 10110111=>1 10111000=>1 10111001=>1 10111010=>3 10111011=>3 10111100=>1 10111101=>1 10111110=>1 10111111=>1 11000000=>0 11000001=>0 11000010=>0 11000011=>0 11000100=>0 11000101=>0 11000110=>3 11000111=>3 11001000=>0 11001001=>0 11001010=>0 11001011=>0 11001100=>0 11001101=>0 11001110=>0 11001111=>0 11010000=>0 11010001=>0 11010010=>0 11010011=>0 11010100=>0 11010101=>0 11010110=>3 11010111=>3 11011000=>2 11011001=>2 11011010=>2 11011011=>2 11011100=>2 11011101=>2 11011110=>2 11011111=>2 11100000=>1 11100001=>1 11100010=>1 11100011=>1 11100100=>1 11100101=>1 11100110=>1 11100111=>1 11101000=>1 11101001=>1 11101010=>1 11101011=>1 11101100=>1 11101101=>1 11101110=>3 11101111=>3 11110000=>1 11110001=>1 11110010=>1 11110011=>1 11110100=>1 11110101=>1 11110110=>1 11110111=>1 11111000=>2 11111001=>2 11111010=>2 11111011=>2 11111100=>2 11111101=>2 11111110=>3 11111111=>3 000000000=>4 000000001=>3 000000010=>3 000000011=>3 000000100=>2 000000101=>2 000000110=>2 000000111=>2 000001000=>2 000001001=>2 000001010=>2 000001011=>2 000001100=>2 000001101=>2 000001110=>2 000001111=>2 000010000=>4 000010001=>1 000010010=>1 000010011=>1 000010100=>1 000010101=>1 000010110=>1 000010111=>1 000011000=>1 000011001=>1 000011010=>1 000011011=>1 000011100=>1 000011101=>1 000011110=>1 000011111=>1 000100000=>3 000100001=>3 000100010=>3 000100011=>3 000100100=>1 000100101=>1 000100110=>1 000100111=>1 000101000=>4 000101001=>1 000101010=>1 000101011=>1 000101100=>1 000101101=>1 000101110=>1 000101111=>1 000110000=>1 000110001=>1 000110010=>1 000110011=>1 000110100=>1 000110101=>1 000110110=>1 000110111=>1 000111000=>4 000111001=>1 000111010=>1 000111011=>1 000111100=>1 000111101=>1 000111110=>1 000111111=>1 001000000=>2 001000001=>2 001000010=>2 001000011=>2 001000100=>4 001000101=>2 001000110=>2 001000111=>2 001001000=>2 001001001=>2 001001010=>2 001001011=>2 001001100=>2 001001101=>2 001001110=>2 001001111=>2 001010000=>3 001010001=>3 001010010=>3 001010011=>3 001010100=>4 001010101=>0 001010110=>0 001010111=>0 001011000=>0 001011001=>0 001011010=>0 001011011=>0 001011100=>0 001011101=>0 001011110=>0 001011111=>0 001100000=>0 001100001=>0 001100010=>0 001100011=>0 001100100=>0 001100101=>0 001100110=>0 001100111=>0 001101000=>0 001101001=>0 001101010=>0 001101011=>0 001101100=>4 001101101=>0 001101110=>0 001101111=>0 001110000=>3 001110001=>3 001110010=>3 001110011=>3 001110100=>0 001110101=>0 001110110=>0 001110111=>0 001111000=>0 001111001=>0 001111010=>0 001111011=>0 001111100=>4 001111101=>0 001111110=>0 001111111=>0 010000000=>1 010000001=>1 010000010=>4 010000011=>1 010000100=>1 010000101=>1 010000110=>1 010000111=>1 010001000=>3 010001001=>3 010001010=>3 010001011=>3 010001100=>1 010001101=>1 010001110=>1 010001111=>1 010010000=>1 010010001=>1 010010010=>4 010010011=>1 010010100=>1 010010101=>1 010010110=>1 010010111=>1 010011000=>1 010011001=>1 010011010=>1 010011011=>1 010011100=>1 010011101=>1 010011110=>1 010011111=>1 010100000=>2 010100001=>2 010100010=>2 010100011=>2 010100100=>2 010100101=>2 010100110=>2 010100111=>2 010101000=>3 010101001=>3 010101010=>4 010101011=>3 010101100=>2 010101101=>2 010101110=>2 010101111=>2 010110000=>1 010110001=>1 010110010=>1 010110011=>1 010110100=>1 010110101=>1 010110110=>1 010110111=>1 010111000=>1 010111001=>1 010111010=>4 010111011=>1 010111100=>1 010111101=>1 010111110=>1 010111111=>1 011000000=>0 011000001=>0 011000010=>0 011000011=>0 011000100=>0 011000101=>0 011000110=>4 011000111=>0 011001000=>0 011001001=>0 011001010=>0 011001011=>0 011001100=>0 011001101=>0 011001110=>0 011001111=>0 011010000=>0 011010001=>0 011010010=>0 011010011=>0 011010100=>0 011010101=>0 011010110=>4 011010111=>0 011011000=>3 011011001=>3 011011010=>3 011011011=>3 011011100=>0 011011101=>0 011011110=>0 011011111=>0 011100000=>2 011100001=>2 011100010=>2 011100011=>2 011100100=>2 011100101=>2 011100110=>2 011100111=>2 011101000=>2 011101001=>2 011101010=>2 011101011=>2 011101100=>2 011101101=>2 011101110=>4 011101111=>2 011110000=>0 011110001=>0 011110010=>0 011110011=>0 011110100=>0 011110101=>0 011110110=>0 011110111=>0 011111000=>3 011111001=>3 011111010=>3 011111011=>3 011111100=>0 011111101=>0 011111110=>4 011111111=>0 100000000=>0 100000001=>4 100000010=>0 100000011=>0 100000100=>3 100000101=>3 100000110=>3 100000111=>3 100001000=>0 100001001=>0 100001010=>0 100001011=>0 100001100=>0 100001101=>0 100001110=>0 100001111=>0 100010000=>2 100010001=>4 100010010=>2 100010011=>2 100010100=>2 100010101=>2 100010110=>2 100010111=>2 100011000=>2 100011001=>2 100011010=>2 100011011=>2 100011100=>2 100011101=>2 100011110=>2 100011111=>2 100100000=>0 100100001=>0 100100010=>0 100100011=>0 100100100=>3 100100101=>3 100100110=>3 100100111=>3 100101000=>0 100101001=>4 100101010=>0 100101011=>0 100101100=>0 100101101=>0 100101110=>0 100101111=>0 100110000=>0 100110001=>0 100110010=>0 100110011=>0 100110100=>0 100110101=>0 100110110=>0 100110111=>0 100111000=>0 100111001=>4 100111010=>0 100111011=>0 100111100=>0 100111101=>0 100111110=>0 100111111=>0 101000000=>1 101000001=>1 101000010=>1 101000011=>1 101000100=>1 101000101=>4 101000110=>1 101000111=>1 101001000=>1 101001001=>1 101001010=>1 101001011=>1 101001100=>1 101001101=>1 101001110=>1 101001111=>1 101010000=>2 101010001=>2 101010010=>2 101010011=>2 101010100=>3 101010101=>4 101010110=>3 101010111=>3 101011000=>2 101011001=>2 101011010=>2 101011011=>2 101011100=>2 101011101=>2 101011110=>2 101011111=>2 101100000=>1 101100001=>1 101100010=>1 101100011=>1 101100100=>1 101100101=>1 101100110=>1 101100111=>1 101101000=>1 101101001=>1 101101010=>1 101101011=>1 101101100=>1 101101101=>4 101101110=>1 101101111=>1 101110000=>1 101110001=>1 101110010=>1 101110011=>1 101110100=>3 101110101=>3 101110110=>3 101110111=>3 101111000=>1 101111001=>1 101111010=>1 101111011=>1 101111100=>1 101111101=>4 101111110=>1 101111111=>1 110000000=>0 110000001=>0 110000010=>0 110000011=>4 110000100=>0 110000101=>0 110000110=>0 110000111=>0 110001000=>0 110001001=>0 110001010=>0 110001011=>0 110001100=>3 110001101=>3 110001110=>3 110001111=>3 110010000=>0 110010001=>0 110010010=>0 110010011=>4 110010100=>0 110010101=>0 110010110=>0 110010111=>0 110011000=>0 110011001=>0 110011010=>0 110011011=>0 110011100=>0 110011101=>0 110011110=>0 110011111=>0 110100000=>0 110100001=>0 110100010=>0 110100011=>0 110100100=>0 110100101=>0 110100110=>0 110100111=>0 110101000=>0 110101001=>0 110101010=>0 110101011=>4 110101100=>3 110101101=>3 110101110=>3 110101111=>3 110110000=>2 110110001=>2 110110010=>2 110110011=>2 110110100=>2 110110101=>2 110110110=>2 110110111=>2 110111000=>2 110111001=>2 110111010=>2 110111011=>4 110111100=>2 110111101=>2 110111110=>2 110111111=>2 111000000=>1 111000001=>1 111000010=>1 111000011=>1 111000100=>1 111000101=>1 111000110=>1 111000111=>4 111001000=>1 111001001=>1 111001010=>1 111001011=>1 111001100=>1 111001101=>1 111001110=>1 111001111=>1 111010000=>1 111010001=>1 111010010=>1 111010011=>1 111010100=>1 111010101=>1 111010110=>1 111010111=>4 111011000=>1 111011001=>1 111011010=>1 111011011=>1 111011100=>3 111011101=>3 111011110=>3 111011111=>3 111100000=>1 111100001=>1 111100010=>1 111100011=>1 111100100=>1 111100101=>1 111100110=>1 111100111=>1 111101000=>1 111101001=>1 111101010=>1 111101011=>1 111101100=>1 111101101=>1 111101110=>1 111101111=>4 111110000=>2 111110001=>2 111110010=>2 111110011=>2 111110100=>2 111110101=>2 111110110=>2 111110111=>2 111111000=>2 111111001=>2 111111010=>2 111111011=>2 111111100=>3 111111101=>3 111111110=>3 111111111=>4
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Description
Half the length of the longest odd length palindromic prefix of a binary word.
More precisely, this statistic is the largest number $k$ such that the word has a palindromic prefix of length $2k+1$.
References
[1] Gabric, D., Shallit, J. Borders, Palindrome Prefixes, and Square Prefixes arXiv:1906.03689
Code
def statistic(w):
    return max((k-1)//2 for k in range(1,len(w)+1,2) if w[:k] == w[:k][::-1])

Created
Jun 11, 2019 at 07:28 by Martin Rubey
Updated
Mar 05, 2020 at 14:09 by Martin Rubey