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Identifier
Values
=>
0=>0 1=>0 00=>0 01=>0 10=>1 11=>0 000=>0 001=>0 010=>1 011=>0 100=>2 101=>1 110=>2 111=>0 0000=>0 0001=>0 0010=>1 0011=>0 0100=>2 0101=>0 0110=>1 0111=>0 1000=>3 1001=>2 1010=>1 1011=>1 1100=>3 1101=>2 1110=>3 1111=>0 00000=>0 00001=>0 00010=>1 00011=>0 00100=>2 00101=>0 00110=>1 00111=>0 01000=>3 01001=>1 01010=>2 01011=>0 01100=>2 01101=>1 01110=>1 01111=>0 10000=>4 10001=>3 10010=>3 10011=>2 10100=>4 10101=>2 10110=>3 10111=>1 11000=>4 11001=>3 11010=>4 11011=>2 11100=>4 11101=>3 11110=>4 11111=>0 000000=>0 000001=>0 000010=>1 000011=>0 000100=>2 000101=>0 000110=>1 000111=>0 001000=>3 001001=>0 001010=>1 001011=>0 001100=>2 001101=>0 001110=>1 001111=>0 010000=>4 010001=>2 010010=>1 010011=>1 010100=>3 010101=>0 010110=>1 010111=>0 011000=>3 011001=>2 011010=>2 011011=>0 011100=>2 011101=>1 011110=>1 011111=>0 100000=>5 100001=>4 100010=>4 100011=>3 100100=>2 100101=>3 100110=>3 100111=>2 101000=>5 101001=>4 101010=>1 101011=>2 101100=>4 101101=>1 101110=>3 101111=>1 110000=>5 110001=>4 110010=>5 110011=>3 110100=>5 110101=>4 110110=>2 110111=>2 111000=>5 111001=>4 111010=>5 111011=>3 111100=>5 111101=>4 111110=>5 111111=>0 0000000=>0 0000001=>0 0000010=>1 0000011=>0 0000100=>2 0000101=>0 0000110=>1 0000111=>0 0001000=>3 0001001=>0 0001010=>1 0001011=>0 0001100=>2 0001101=>0 0001110=>1 0001111=>0 0010000=>4 0010001=>1 0010010=>2 0010011=>0 0010100=>2 0010101=>0 0010110=>1 0010111=>0 0011000=>3 0011001=>1 0011010=>1 0011011=>0 0011100=>2 0011101=>0 0011110=>1 0011111=>0 0100000=>5 0100001=>3 0100010=>3 0100011=>2 0100100=>4 0100101=>1 0100110=>2 0100111=>1 0101000=>4 0101001=>2 0101010=>3 0101011=>0 0101100=>2 0101101=>1 0101110=>1 0101111=>0 0110000=>4 0110001=>3 0110010=>3 0110011=>1 0110100=>3 0110101=>2 0110110=>2 0110111=>0 0111000=>3 0111001=>2 0111010=>2 0111011=>1 0111100=>2 0111101=>1 0111110=>1 0111111=>0 1000000=>6 1000001=>5 1000010=>5 1000011=>4 1000100=>5 1000101=>4 1000110=>4 1000111=>3 1001000=>6 1001001=>4 1001010=>4 1001011=>3 1001100=>5 1001101=>3 1001110=>3 1001111=>2 1010000=>6 1010001=>5 1010010=>5 1010011=>4 1010100=>6 1010101=>3 1010110=>4 1010111=>2 1011000=>5 1011001=>4 1011010=>5 1011011=>2 1011100=>4 1011101=>3 1011110=>3 1011111=>1 1100000=>6 1100001=>5 1100010=>6 1100011=>4 1100100=>6 1100101=>5 1100110=>5 1100111=>3 1101000=>6 1101001=>5 1101010=>6 1101011=>4 1101100=>6 1101101=>4 1101110=>5 1101111=>2 1110000=>6 1110001=>5 1110010=>6 1110011=>4 1110100=>6 1110101=>5 1110110=>6 1110111=>3 1111000=>6 1111001=>5 1111010=>6 1111011=>4 1111100=>6 1111101=>5 1111110=>6 1111111=>0 00000000=>0 00000001=>0 00000010=>1 00000011=>0 00000100=>2 00000101=>0 00000110=>1 00000111=>0 00001000=>3 00001001=>0 00001010=>1 00001011=>0 00001100=>2 00001101=>0 00001110=>1 00001111=>0 00010000=>4 00010001=>0 00010010=>1 00010011=>0 00010100=>2 00010101=>0 00010110=>1 00010111=>0 00011000=>3 00011001=>0 00011010=>1 00011011=>0 00011100=>2 00011101=>0 00011110=>1 00011111=>0 00100000=>5 00100001=>2 00100010=>1 00100011=>1 00100100=>3 00100101=>0 00100110=>1 00100111=>0 00101000=>3 00101001=>1 00101010=>1 00101011=>0 00101100=>2 00101101=>0 00101110=>1 00101111=>0 00110000=>4 00110001=>2 00110010=>2 00110011=>0 00110100=>2 00110101=>0 00110110=>1 00110111=>0 00111000=>3 00111001=>1 00111010=>1 00111011=>0 00111100=>2 00111101=>0 00111110=>1 00111111=>0 01000000=>6 01000001=>4 01000010=>4 01000011=>3 01000100=>2 01000101=>2 01000110=>3 01000111=>2 01001000=>5 01001001=>2 01001010=>3 01001011=>1 01001100=>3 01001101=>1 01001110=>2 01001111=>1 01010000=>5 01010001=>3 01010010=>4 01010011=>2 01010100=>4 01010101=>0 01010110=>1 01010111=>0 01011000=>3 01011001=>2 01011010=>2 01011011=>0 01011100=>2 01011101=>1 01011110=>1 01011111=>0 01100000=>5 01100001=>4 01100010=>4 01100011=>2 01100100=>4 01100101=>3 01100110=>1 01100111=>1 01101000=>4 01101001=>3 01101010=>3 01101011=>1 01101100=>3 01101101=>2 01101110=>1 01101111=>0 01110000=>4 01110001=>3 01110010=>3 01110011=>2 01110100=>3 01110101=>2 01110110=>2 01110111=>0 01111000=>3 01111001=>2 01111010=>2 01111011=>1 01111100=>2 01111101=>1 01111110=>1 01111111=>0 10000000=>7 10000001=>6 10000010=>6 10000011=>5 10000100=>6 10000101=>5 10000110=>5 10000111=>4 10001000=>3 10001001=>5 10001010=>5 10001011=>4 10001100=>5 10001101=>4 10001110=>4 10001111=>3 10010000=>7 10010001=>6 10010010=>5 10010011=>4 10010100=>6 10010101=>4 10010110=>4 10010111=>3 10011000=>6 10011001=>2 10011010=>4 10011011=>3 10011100=>5 10011101=>3 10011110=>3 10011111=>2 10100000=>7 10100001=>6 10100010=>6 10100011=>5 10100100=>7 10100101=>5 10100110=>5 10100111=>4 10101000=>7 10101001=>6 10101010=>1 10101011=>3 10101100=>5 10101101=>3 10101110=>4 10101111=>2 10110000=>6 10110001=>5 10110010=>6 10110011=>4 10110100=>6 10110101=>4 10110110=>5 10110111=>2 10111000=>5 10111001=>4 10111010=>5 10111011=>1 10111100=>4 10111101=>3 10111110=>3 10111111=>1 11000000=>7 11000001=>6 11000010=>7 11000011=>5 11000100=>7 11000101=>6 11000110=>6 11000111=>4 11001000=>7 11001001=>6 11001010=>7 11001011=>5 11001100=>3 11001101=>5 11001110=>5 11001111=>3 11010000=>7 11010001=>6 11010010=>7 11010011=>5 11010100=>7 11010101=>6 11010110=>6 11010111=>4 11011000=>7 11011001=>6 11011010=>7 11011011=>4 11011100=>6 11011101=>2 11011110=>5 11011111=>2 11100000=>7 11100001=>6 11100010=>7 11100011=>5 11100100=>7 11100101=>6 11100110=>7 11100111=>4 11101000=>7 11101001=>6 11101010=>7 11101011=>5 11101100=>7 11101101=>6 11101110=>3 11101111=>3 11110000=>7 11110001=>6 11110010=>7 11110011=>5 11110100=>7 11110101=>6 11110110=>7 11110111=>4 11111000=>7 11111001=>6 11111010=>7 11111011=>5 11111100=>7 11111101=>6 11111110=>7 11111111=>0 000000000=>0 000000001=>0 000000010=>1 000000011=>0 000000100=>2 000000101=>0 000000110=>1 000000111=>0 000001000=>3 000001001=>0 000001010=>1 000001011=>0 000001100=>2 000001101=>0 000001110=>1 000001111=>0 000010000=>4 000010001=>0 000010010=>1 000010011=>0 000010100=>2 000010101=>0 000010110=>1 000010111=>0 000011000=>3 000011001=>0 000011010=>1 000011011=>0 000011100=>2 000011101=>0 000011110=>1 000011111=>0 000100000=>5 000100001=>1 000100010=>2 000100011=>0 000100100=>2 000100101=>0 000100110=>1 000100111=>0 000101000=>3 000101001=>0 000101010=>1 000101011=>0 000101100=>2 000101101=>0 000101110=>1 000101111=>0 000110000=>4 000110001=>1 000110010=>1 000110011=>0 000110100=>2 000110101=>0 000110110=>1 000110111=>0 000111000=>3 000111001=>0 000111010=>1 000111011=>0 000111100=>2 000111101=>0 000111110=>1 000111111=>0 001000000=>6 001000001=>3 001000010=>3 001000011=>2 001000100=>4 001000101=>1 001000110=>2 001000111=>1 001001000=>4 001001001=>0 001001010=>1 001001011=>0 001001100=>2 001001101=>0 001001110=>1 001001111=>0 001010000=>4 001010001=>2 001010010=>2 001010011=>0 001010100=>2 001010101=>0 001010110=>1 001010111=>0 001011000=>3 001011001=>1 001011010=>1 001011011=>0 001011100=>2 001011101=>0 001011110=>1 001011111=>0 001100000=>5 001100001=>3 001100010=>3 001100011=>1 001100100=>3 001100101=>1 001100110=>2 001100111=>0 001101000=>3 001101001=>1 001101010=>1 001101011=>0 001101100=>2 001101101=>0 001101110=>1 001101111=>0 001110000=>4 001110001=>2 001110010=>2 001110011=>1 001110100=>2 001110101=>0 001110110=>1 001110111=>0 001111000=>3 001111001=>1 001111010=>1 001111011=>0 001111100=>2 001111101=>0 001111110=>1 001111111=>0 010000000=>7 010000001=>5 010000010=>5 010000011=>4 010000100=>5 010000101=>3 010000110=>4 010000111=>3 010001000=>6 010001001=>3 010001010=>3 010001011=>2 010001100=>4 010001101=>2 010001110=>3 010001111=>2 010010000=>6 010010001=>4 010010010=>1 010010011=>2 010010100=>4 010010101=>1 010010110=>2 010010111=>1 010011000=>4 010011001=>2 010011010=>3 010011011=>1 010011100=>3 010011101=>1 010011110=>2 010011111=>1 010100000=>6 010100001=>4 010100010=>5 010100011=>3 010100100=>5 010100101=>2 010100110=>3 010100111=>2 010101000=>5 010101001=>3 010101010=>4 010101011=>0 010101100=>2 010101101=>1 010101110=>1 010101111=>0 010110000=>4 010110001=>3 010110010=>3 010110011=>1 010110100=>3 010110101=>2 010110110=>1 010110111=>0 010111000=>3 010111001=>2 010111010=>2 010111011=>0 010111100=>2 010111101=>1 010111110=>1 010111111=>0 011000000=>6 011000001=>5 011000010=>5 011000011=>3 011000100=>5 011000101=>4 011000110=>3 011000111=>2 011001000=>5 011001001=>4 011001010=>4 011001011=>2 011001100=>4 011001101=>2 011001110=>2 011001111=>1 011010000=>5 011010001=>4 011010010=>4 011010011=>2 011010100=>4 011010101=>3 011010110=>3 011010111=>1 011011000=>4 011011001=>3 011011010=>3 011011011=>0 011011100=>2 011011101=>1 011011110=>1 011011111=>0 011100000=>5 011100001=>4 011100010=>4 011100011=>3 011100100=>4 011100101=>3 011100110=>3 011100111=>1 011101000=>4 011101001=>3 011101010=>3 011101011=>2 011101100=>3 011101101=>2 011101110=>2 011101111=>0 011110000=>4 011110001=>3 011110010=>3 011110011=>2 011110100=>3 011110101=>2 011110110=>2 011110111=>1 011111000=>3 011111001=>2 011111010=>2 011111011=>1 011111100=>2 011111101=>1 011111110=>1 011111111=>0 100000000=>8 100000001=>7 100000010=>7 100000011=>6 100000100=>7 100000101=>6 100000110=>6 100000111=>5 100001000=>7 100001001=>6 100001010=>6 100001011=>5 100001100=>6 100001101=>5 100001110=>5 100001111=>4 100010000=>8 100010001=>6 100010010=>6 100010011=>5 100010100=>6 100010101=>5 100010110=>5 100010111=>4 100011000=>7 100011001=>5 100011010=>5 100011011=>4 100011100=>5 100011101=>4 100011110=>4 100011111=>3 100100000=>8 100100001=>7 100100010=>7 100100011=>6 100100100=>2 100100101=>5 100100110=>5 100100111=>4 100101000=>7 100101001=>5 100101010=>5 100101011=>4 100101100=>6 100101101=>4 100101110=>4 100101111=>3 100110000=>7 100110001=>6 100110010=>6 100110011=>4 100110100=>6 100110101=>4 100110110=>4 100110111=>3 100111000=>6 100111001=>5 100111010=>4 100111011=>3 100111100=>5 100111101=>3 100111110=>3 100111111=>2 101000000=>8 101000001=>7 101000010=>7 101000011=>6 101000100=>8 101000101=>6 101000110=>6 101000111=>5 101001000=>8 101001001=>7 101001010=>6 101001011=>5 101001100=>7 101001101=>5 101001110=>5 101001111=>4 101010000=>8 101010001=>7 101010010=>7 101010011=>6 101010100=>8 101010101=>4 101010110=>5 101010111=>3 101011000=>6 101011001=>5 101011010=>6 101011011=>3 101011100=>5 101011101=>3 101011110=>4 101011111=>2 101100000=>7 101100001=>6 101100010=>7 101100011=>5 101100100=>7 101100101=>5 101100110=>6 101100111=>4 101101000=>7 101101001=>6 101101010=>7 101101011=>4 101101100=>6 101101101=>1 101101110=>4 101101111=>2 101110000=>6 101110001=>5 101110010=>6 101110011=>4 101110100=>6 101110101=>5 101110110=>5 101110111=>2 101111000=>5 101111001=>4 101111010=>5 101111011=>3 101111100=>4 101111101=>3 101111110=>3 101111111=>1 110000000=>8 110000001=>7 110000010=>8 110000011=>6 110000100=>8 110000101=>7 110000110=>7 110000111=>5 110001000=>8 110001001=>7 110001010=>8 110001011=>6 110001100=>7 110001101=>6 110001110=>6 110001111=>4 110010000=>8 110010001=>7 110010010=>8 110010011=>6 110010100=>8 110010101=>7 110010110=>7 110010111=>5 110011000=>8 110011001=>6 110011010=>7 110011011=>5 110011100=>7 110011101=>5 110011110=>5 110011111=>3 110100000=>8 110100001=>7 110100010=>8 110100011=>6 110100100=>8 110100101=>7 110100110=>7 110100111=>5 110101000=>8 110101001=>7 110101010=>8 110101011=>6 110101100=>8 110101101=>6 110101110=>6 110101111=>4 110110000=>8 110110001=>7 110110010=>8 110110011=>6 110110100=>8 110110101=>7 110110110=>2 110110111=>4 110111000=>7 110111001=>6 110111010=>7 110111011=>4 110111100=>6 110111101=>5 110111110=>5 110111111=>2 111000000=>8 111000001=>7 111000010=>8 111000011=>6 111000100=>8 111000101=>7 111000110=>8 111000111=>5 111001000=>8 111001001=>7 111001010=>8 111001011=>6 111001100=>8 111001101=>7 111001110=>7 111001111=>4 111010000=>8 111010001=>7 111010010=>8 111010011=>6 111010100=>8 111010101=>7 111010110=>8 111010111=>5 111011000=>8 111011001=>7 111011010=>8 111011011=>6 111011100=>8 111011101=>6 111011110=>7 111011111=>3 111100000=>8 111100001=>7 111100010=>8 111100011=>6 111100100=>8 111100101=>7 111100110=>8 111100111=>5 111101000=>8 111101001=>7 111101010=>8 111101011=>6 111101100=>8 111101101=>7 111101110=>8 111101111=>4 111110000=>8 111110001=>7 111110010=>8 111110011=>6 111110100=>8 111110101=>7 111110110=>8 111110111=>5 111111000=>8 111111001=>7 111111010=>8 111111011=>6 111111100=>8 111111101=>7 111111110=>8 111111111=>0
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Description
The index of a given binary word in the lex-order among all its cyclic shifts.
References
[1] Ahlbach, C., Swanson, J. P. Refined cyclic sieving on words for the major index statistic MathSciNet:3836732
Code
def statistic(w):
    orbit = [ tuple(w[i:] + w[:i]) for i in range(len(w)) ]
    return sorted(set(orbit)).index(tuple(w))

Created
Jul 02, 2019 at 10:59 by Christian Stump
Updated
Jul 02, 2019 at 10:59 by Christian Stump