edit this statistic or download as text // json
Identifier
Values
=>
0=>0 1=>0 00=>0 01=>0 10=>0 11=>0 000=>0 001=>0 010=>0 011=>0 100=>1 101=>0 110=>1 111=>0 0000=>0 0001=>0 0010=>0 0011=>0 0100=>1 0101=>0 0110=>1 0111=>0 1000=>2 1001=>1 1010=>2 1011=>0 1100=>0 1101=>1 1110=>2 1111=>0 00000=>0 00001=>0 00010=>0 00011=>0 00100=>1 00101=>0 00110=>1 00111=>0 01000=>2 01001=>1 01010=>2 01011=>0 01100=>0 01101=>1 01110=>2 01111=>0 10000=>3 10001=>2 10010=>3 10011=>1 10100=>1 10101=>2 10110=>3 10111=>0 11000=>2 11001=>0 11010=>1 11011=>1 11100=>2 11101=>2 11110=>3 11111=>0 000000=>0 000001=>0 000010=>0 000011=>0 000100=>1 000101=>0 000110=>1 000111=>0 001000=>2 001001=>1 001010=>2 001011=>0 001100=>0 001101=>1 001110=>2 001111=>0 010000=>3 010001=>2 010010=>3 010011=>1 010100=>1 010101=>2 010110=>3 010111=>0 011000=>2 011001=>0 011010=>1 011011=>1 011100=>2 011101=>2 011110=>3 011111=>0 100000=>4 100001=>3 100010=>4 100011=>2 100100=>2 100101=>3 100110=>4 100111=>1 101000=>3 101001=>1 101010=>2 101011=>2 101100=>3 101101=>3 101110=>4 101111=>0 110000=>4 110001=>2 110010=>3 110011=>0 110100=>4 110101=>1 110110=>2 110111=>1 111000=>0 111001=>2 111010=>3 111011=>2 111100=>4 111101=>3 111110=>4 111111=>0 0000000=>0 0000001=>0 0000010=>0 0000011=>0 0000100=>1 0000101=>0 0000110=>1 0000111=>0 0001000=>2 0001001=>1 0001010=>2 0001011=>0 0001100=>0 0001101=>1 0001110=>2 0001111=>0 0010000=>3 0010001=>2 0010010=>3 0010011=>1 0010100=>1 0010101=>2 0010110=>3 0010111=>0 0011000=>2 0011001=>0 0011010=>1 0011011=>1 0011100=>2 0011101=>2 0011110=>3 0011111=>0 0100000=>4 0100001=>3 0100010=>4 0100011=>2 0100100=>2 0100101=>3 0100110=>4 0100111=>1 0101000=>3 0101001=>1 0101010=>2 0101011=>2 0101100=>3 0101101=>3 0101110=>4 0101111=>0 0110000=>4 0110001=>2 0110010=>3 0110011=>0 0110100=>4 0110101=>1 0110110=>2 0110111=>1 0111000=>0 0111001=>2 0111010=>3 0111011=>2 0111100=>4 0111101=>3 0111110=>4 0111111=>0 1000000=>5 1000001=>4 1000010=>5 1000011=>3 1000100=>3 1000101=>4 1000110=>5 1000111=>2 1001000=>4 1001001=>2 1001010=>3 1001011=>3 1001100=>4 1001101=>4 1001110=>5 1001111=>1 1010000=>5 1010001=>3 1010010=>4 1010011=>1 1010100=>5 1010101=>2 1010110=>3 1010111=>2 1011000=>1 1011001=>3 1011010=>4 1011011=>3 1011100=>5 1011101=>4 1011110=>5 1011111=>0 1100000=>6 1100001=>4 1100010=>5 1100011=>2 1100100=>6 1100101=>3 1100110=>4 1100111=>0 1101000=>2 1101001=>4 1101010=>5 1101011=>1 1101100=>6 1101101=>2 1101110=>3 1101111=>1 1110000=>3 1110001=>0 1110010=>1 1110011=>2 1110100=>2 1110101=>3 1110110=>4 1110111=>2 1111000=>3 1111001=>4 1111010=>5 1111011=>3 1111100=>6 1111101=>4 1111110=>5 1111111=>0 00000000=>0 00000001=>0 00000010=>0 00000011=>0 00000100=>1 00000101=>0 00000110=>1 00000111=>0 00001000=>2 00001001=>1 00001010=>2 00001011=>0 00001100=>0 00001101=>1 00001110=>2 00001111=>0 00010000=>3 00010001=>2 00010010=>3 00010011=>1 00010100=>1 00010101=>2 00010110=>3 00010111=>0 00011000=>2 00011001=>0 00011010=>1 00011011=>1 00011100=>2 00011101=>2 00011110=>3 00011111=>0 00100000=>4 00100001=>3 00100010=>4 00100011=>2 00100100=>2 00100101=>3 00100110=>4 00100111=>1 00101000=>3 00101001=>1 00101010=>2 00101011=>2 00101100=>3 00101101=>3 00101110=>4 00101111=>0 00110000=>4 00110001=>2 00110010=>3 00110011=>0 00110100=>4 00110101=>1 00110110=>2 00110111=>1 00111000=>0 00111001=>2 00111010=>3 00111011=>2 00111100=>4 00111101=>3 00111110=>4 00111111=>0 01000000=>5 01000001=>4 01000010=>5 01000011=>3 01000100=>3 01000101=>4 01000110=>5 01000111=>2 01001000=>4 01001001=>2 01001010=>3 01001011=>3 01001100=>4 01001101=>4 01001110=>5 01001111=>1 01010000=>5 01010001=>3 01010010=>4 01010011=>1 01010100=>5 01010101=>2 01010110=>3 01010111=>2 01011000=>1 01011001=>3 01011010=>4 01011011=>3 01011100=>5 01011101=>4 01011110=>5 01011111=>0 01100000=>6 01100001=>4 01100010=>5 01100011=>2 01100100=>6 01100101=>3 01100110=>4 01100111=>0 01101000=>2 01101001=>4 01101010=>5 01101011=>1 01101100=>6 01101101=>2 01101110=>3 01101111=>1 01110000=>3 01110001=>0 01110010=>1 01110011=>2 01110100=>2 01110101=>3 01110110=>4 01110111=>2 01111000=>3 01111001=>4 01111010=>5 01111011=>3 01111100=>6 01111101=>4 01111110=>5 01111111=>0 10000000=>6 10000001=>5 10000010=>6 10000011=>4 10000100=>4 10000101=>5 10000110=>6 10000111=>3 10001000=>5 10001001=>3 10001010=>4 10001011=>4 10001100=>5 10001101=>5 10001110=>6 10001111=>2 10010000=>6 10010001=>4 10010010=>5 10010011=>2 10010100=>6 10010101=>3 10010110=>4 10010111=>3 10011000=>2 10011001=>4 10011010=>5 10011011=>4 10011100=>6 10011101=>5 10011110=>6 10011111=>1 10100000=>7 10100001=>5 10100010=>6 10100011=>3 10100100=>7 10100101=>4 10100110=>5 10100111=>1 10101000=>3 10101001=>5 10101010=>6 10101011=>2 10101100=>7 10101101=>3 10101110=>4 10101111=>2 10110000=>4 10110001=>1 10110010=>2 10110011=>3 10110100=>3 10110101=>4 10110110=>5 10110111=>3 10111000=>4 10111001=>5 10111010=>6 10111011=>4 10111100=>7 10111101=>5 10111110=>6 10111111=>0 11000000=>8 11000001=>6 11000010=>7 11000011=>4 11000100=>8 11000101=>5 11000110=>6 11000111=>2 11001000=>4 11001001=>6 11001010=>7 11001011=>3 11001100=>8 11001101=>4 11001110=>5 11001111=>0 11010000=>5 11010001=>2 11010010=>3 11010011=>4 11010100=>4 11010101=>5 11010110=>6 11010111=>1 11011000=>5 11011001=>6 11011010=>7 11011011=>2 11011100=>8 11011101=>3 11011110=>4 11011111=>1 11100000=>6 11100001=>3 11100010=>4 11100011=>0 11100100=>5 11100101=>1 11100110=>2 11100111=>2 11101000=>6 11101001=>2 11101010=>3 11101011=>3 11101100=>4 11101101=>4 11101110=>5 11101111=>2 11110000=>0 11110001=>3 11110010=>4 11110011=>4 11110100=>5 11110101=>5 11110110=>6 11110111=>3 11111000=>6 11111001=>6 11111010=>7 11111011=>4 11111100=>8 11111101=>5 11111110=>6 11111111=>0 000000000=>0 000000001=>0 000000010=>0 000000011=>0 000000100=>1 000000101=>0 000000110=>1 000000111=>0 000001000=>2 000001001=>1 000001010=>2 000001011=>0 000001100=>0 000001101=>1 000001110=>2 000001111=>0 000010000=>3 000010001=>2 000010010=>3 000010011=>1 000010100=>1 000010101=>2 000010110=>3 000010111=>0 000011000=>2 000011001=>0 000011010=>1 000011011=>1 000011100=>2 000011101=>2 000011110=>3 000011111=>0 000100000=>4 000100001=>3 000100010=>4 000100011=>2 000100100=>2 000100101=>3 000100110=>4 000100111=>1 000101000=>3 000101001=>1 000101010=>2 000101011=>2 000101100=>3 000101101=>3 000101110=>4 000101111=>0 000110000=>4 000110001=>2 000110010=>3 000110011=>0 000110100=>4 000110101=>1 000110110=>2 000110111=>1 000111000=>0 000111001=>2 000111010=>3 000111011=>2 000111100=>4 000111101=>3 000111110=>4 000111111=>0 001000000=>5 001000001=>4 001000010=>5 001000011=>3 001000100=>3 001000101=>4 001000110=>5 001000111=>2 001001000=>4 001001001=>2 001001010=>3 001001011=>3 001001100=>4 001001101=>4 001001110=>5 001001111=>1 001010000=>5 001010001=>3 001010010=>4 001010011=>1 001010100=>5 001010101=>2 001010110=>3 001010111=>2 001011000=>1 001011001=>3 001011010=>4 001011011=>3 001011100=>5 001011101=>4 001011110=>5 001011111=>0 001100000=>6 001100001=>4 001100010=>5 001100011=>2 001100100=>6 001100101=>3 001100110=>4 001100111=>0 001101000=>2 001101001=>4 001101010=>5 001101011=>1 001101100=>6 001101101=>2 001101110=>3 001101111=>1 001110000=>3 001110001=>0 001110010=>1 001110011=>2 001110100=>2 001110101=>3 001110110=>4 001110111=>2 001111000=>3 001111001=>4 001111010=>5 001111011=>3 001111100=>6 001111101=>4 001111110=>5 001111111=>0 010000000=>6 010000001=>5 010000010=>6 010000011=>4 010000100=>4 010000101=>5 010000110=>6 010000111=>3 010001000=>5 010001001=>3 010001010=>4 010001011=>4 010001100=>5 010001101=>5 010001110=>6 010001111=>2 010010000=>6 010010001=>4 010010010=>5 010010011=>2 010010100=>6 010010101=>3 010010110=>4 010010111=>3 010011000=>2 010011001=>4 010011010=>5 010011011=>4 010011100=>6 010011101=>5 010011110=>6 010011111=>1 010100000=>7 010100001=>5 010100010=>6 010100011=>3 010100100=>7 010100101=>4 010100110=>5 010100111=>1 010101000=>3 010101001=>5 010101010=>6 010101011=>2 010101100=>7 010101101=>3 010101110=>4 010101111=>2 010110000=>4 010110001=>1 010110010=>2 010110011=>3 010110100=>3 010110101=>4 010110110=>5 010110111=>3 010111000=>4 010111001=>5 010111010=>6 010111011=>4 010111100=>7 010111101=>5 010111110=>6 010111111=>0 011000000=>8 011000001=>6 011000010=>7 011000011=>4 011000100=>8 011000101=>5 011000110=>6 011000111=>2 011001000=>4 011001001=>6 011001010=>7 011001011=>3 011001100=>8 011001101=>4 011001110=>5 011001111=>0 011010000=>5 011010001=>2 011010010=>3 011010011=>4 011010100=>4 011010101=>5 011010110=>6 011010111=>1 011011000=>5 011011001=>6 011011010=>7 011011011=>2 011011100=>8 011011101=>3 011011110=>4 011011111=>1 011100000=>6 011100001=>3 011100010=>4 011100011=>0 011100100=>5 011100101=>1 011100110=>2 011100111=>2 011101000=>6 011101001=>2 011101010=>3 011101011=>3 011101100=>4 011101101=>4 011101110=>5 011101111=>2 011110000=>0 011110001=>3 011110010=>4 011110011=>4 011110100=>5 011110101=>5 011110110=>6 011110111=>3 011111000=>6 011111001=>6 011111010=>7 011111011=>4 011111100=>8 011111101=>5 011111110=>6 011111111=>0 100000000=>7 100000001=>6 100000010=>7 100000011=>5 100000100=>5 100000101=>6 100000110=>7 100000111=>4 100001000=>6 100001001=>4 100001010=>5 100001011=>5 100001100=>6 100001101=>6 100001110=>7 100001111=>3 100010000=>7 100010001=>5 100010010=>6 100010011=>3 100010100=>7 100010101=>4 100010110=>5 100010111=>4 100011000=>3 100011001=>5 100011010=>6 100011011=>5 100011100=>7 100011101=>6 100011110=>7 100011111=>2 100100000=>8 100100001=>6 100100010=>7 100100011=>4 100100100=>8 100100101=>5 100100110=>6 100100111=>2 100101000=>4 100101001=>6 100101010=>7 100101011=>3 100101100=>8 100101101=>4 100101110=>5 100101111=>3 100110000=>5 100110001=>2 100110010=>3 100110011=>4 100110100=>4 100110101=>5 100110110=>6 100110111=>4 100111000=>5 100111001=>6 100111010=>7 100111011=>5 100111100=>8 100111101=>6 100111110=>7 100111111=>1 101000000=>9 101000001=>7 101000010=>8 101000011=>5 101000100=>9 101000101=>6 101000110=>7 101000111=>3 101001000=>5 101001001=>7 101001010=>8 101001011=>4 101001100=>9 101001101=>5 101001110=>6 101001111=>1 101010000=>6 101010001=>3 101010010=>4 101010011=>5 101010100=>5 101010101=>6 101010110=>7 101010111=>2 101011000=>6 101011001=>7 101011010=>8 101011011=>3 101011100=>9 101011101=>4 101011110=>5 101011111=>2 101100000=>7 101100001=>4 101100010=>5 101100011=>1 101100100=>6 101100101=>2 101100110=>3 101100111=>3 101101000=>7 101101001=>3 101101010=>4 101101011=>4 101101100=>5 101101101=>5 101101110=>6 101101111=>3 101110000=>1 101110001=>4 101110010=>5 101110011=>5 101110100=>6 101110101=>6 101110110=>7 101110111=>4 101111000=>7 101111001=>7 101111010=>8 101111011=>5 101111100=>9 101111101=>6 101111110=>7 101111111=>0 110000000=>10 110000001=>8 110000010=>9 110000011=>6 110000100=>10 110000101=>7 110000110=>8 110000111=>4 110001000=>6 110001001=>8 110001010=>9 110001011=>5 110001100=>10 110001101=>6 110001110=>7 110001111=>2 110010000=>7 110010001=>4 110010010=>5 110010011=>6 110010100=>6 110010101=>7 110010110=>8 110010111=>3 110011000=>7 110011001=>8 110011010=>9 110011011=>4 110011100=>10 110011101=>5 110011110=>6 110011111=>0 110100000=>8 110100001=>5 110100010=>6 110100011=>2 110100100=>7 110100101=>3 110100110=>4 110100111=>4 110101000=>8 110101001=>4 110101010=>5 110101011=>5 110101100=>6 110101101=>6 110101110=>7 110101111=>1 110110000=>2 110110001=>5 110110010=>6 110110011=>6 110110100=>7 110110101=>7 110110110=>8 110110111=>2 110111000=>8 110111001=>8 110111010=>9 110111011=>3 110111100=>10 110111101=>4 110111110=>5 110111111=>1 111000000=>9 111000001=>6 111000010=>7 111000011=>3 111000100=>8 111000101=>4 111000110=>5 111000111=>0 111001000=>9 111001001=>5 111001010=>6 111001011=>1 111001100=>7 111001101=>2 111001110=>3 111001111=>2 111010000=>3 111010001=>6 111010010=>7 111010011=>2 111010100=>8 111010101=>3 111010110=>4 111010111=>3 111011000=>9 111011001=>4 111011010=>5 111011011=>4 111011100=>6 111011101=>5 111011110=>6 111011111=>2 111100000=>4 111100001=>0 111100010=>1 111100011=>3 111100100=>2 111100101=>4 111100110=>5 111100111=>4 111101000=>3 111101001=>5 111101010=>6 111101011=>5 111101100=>7 111101101=>6 111101110=>7 111101111=>3 111110000=>4 111110001=>6 111110010=>7 111110011=>6 111110100=>8 111110101=>7 111110110=>8 111110111=>4 111111000=>9 111111001=>8 111111010=>9 111111011=>5 111111100=>10 111111101=>6 111111110=>7 111111111=>0 1010101010=>6 1010101100=>7 1010110010=>7 1010110100=>8 1010111000=>9 1011001010=>7 1011001100=>8 1011010010=>8 1011010100=>9 1011011000=>10 1011100010=>2 1011100100=>3 1011101000=>4 1011110000=>5 1100101010=>7 1100101100=>8 1100110010=>8 1100110100=>9 1100111000=>10 1101001010=>8 1101001100=>9 1101010010=>9 1101010100=>10 1101011000=>11 1101100010=>3 1101100100=>4 1101101000=>5 1101110000=>6 1110001010=>9 1110001100=>10 1110010010=>10 1110010100=>11 1110011000=>12 1110100010=>4 1110100100=>5 1110101000=>6 1110110000=>7 1111000010=>5 1111000100=>6 1111001000=>7 1111010000=>8 1111100000=>0 1111111110=>8 1111111101=>7 1111000001=>4 1110111110=>7 1110100001=>3 1110011110=>4 1110010001=>9 1110001001=>8 1110000101=>7 1101111110=>6 1101100001=>2 1101010001=>8 1101001110=>5 1101001001=>7 1101000101=>6 1100110001=>7 1100101001=>6 1100100110=>6 1100100101=>5 1100011001=>10 1100010101=>9 1100010010=>7 1100001000=>8 1011111110=>8 1011111101=>7 1011100001=>1 1011010001=>7 1011001001=>6 1011000101=>5 1010111110=>6 1010110001=>6 1010101001=>5 1010100110=>5 1010100101=>4 1010011010=>10 1010011001=>9 1010010110=>9 1010010101=>8 1010001110=>8 1001111110=>8 1001111000=>8 1001110100=>7 1001110001=>5 1001101100=>6 1001101010=>5 1001101001=>4 1001100110=>4 1001100101=>3 1001011110=>6 1001011010=>9 1001011001=>8 1001010110=>8 1001010101=>7 1001001110=>7 1000111110=>8 1000111100=>9 1000111010=>8 1000110110=>7 1000101110=>6 1000011110=>8 1000010110=>6 1000001110=>8 1000001010=>6 1000000110=>8 1000000100=>6 1000000010=>8 1000000000=>8 0111111111=>0 0111111101=>6 0111111001=>8 0111110001=>6 0111100001=>0 0111000001=>6 0110000001=>8 0100000001=>6 0000000001=>0 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101111010000=>5 101111100000=>6 110010101010=>13 110010101100=>14 110010110010=>14 110010110100=>15 110010111000=>16 110011001010=>7 110011001100=>8 110011010010=>8 110011010100=>9 110011011000=>10 110011100010=>9 110011100100=>10 110011101000=>11 110011110000=>12 110100101010=>14 110100101100=>15 110100110010=>15 110100110100=>16 110100111000=>17 110101001010=>8 110101001100=>9 110101010010=>9 110101010100=>10 110101011000=>11 110101100010=>10 110101100100=>11 110101101000=>12 110101110000=>13 110110001010=>9 110110001100=>10 110110010010=>10 110110010100=>11 110110011000=>12 110110100010=>11 110110100100=>12 110110101000=>13 110110110000=>14 110111000010=>3 110111000100=>4 110111001000=>5 110111010000=>6 110111100000=>7 111000101010=>15 111000101100=>16 111000110010=>16 111000110100=>17 111000111000=>18 111001001010=>9 111001001100=>10 111001010010=>10 111001010100=>11 111001011000=>12 111001100010=>11 111001100100=>12 111001101000=>13 111001110000=>14 111010001010=>10 111010001100=>11 111010010010=>11 111010010100=>12 111010011000=>13 111010100010=>12 111010100100=>13 111010101000=>14 111010110000=>15 111011000010=>4 111011000100=>5 111011001000=>6 111011010000=>7 111011100000=>8 111100001010=>11 111100001100=>12 111100010010=>12 111100010100=>13 111100011000=>14 111100100010=>13 111100100100=>14 111100101000=>15 111100110000=>16 111101000010=>5 111101000100=>6 111101001000=>7 111101010000=>8 111101100000=>9 111110000010=>6 111110000100=>7 111110001000=>8 111110010000=>9 111110100000=>10 111111000000=>0 100001111100=>12 010101010101=>6 010101010011=>5 010101001101=>5 010101001011=>4 010101000111=>3 010100110101=>10 010100110011=>9 010100101101=>9 010100101011=>8 010100100111=>7 010100011101=>8 010100011011=>7 010100010111=>6 010100001111=>5 010011010101=>5 010011010011=>4 010011001101=>4 010011001011=>3 010011000111=>2 010010110101=>9 010010110011=>8 010010101101=>8 010010101011=>7 010010100111=>6 010010011101=>7 010010011011=>6 010010010111=>5 010010001111=>4 010001110101=>8 010001110011=>7 010001101101=>7 010001101011=>6 010001100111=>5 010001011101=>6 010001011011=>5 010001010111=>4 010001001111=>3 010000111101=>8 010000111011=>7 010000110111=>6 010000101111=>5 010000011111=>4 001101010101=>5 001101010011=>4 001101001101=>4 001101001011=>3 001101000111=>2 001100110101=>9 001100110011=>8 001100101101=>8 001100101011=>7 001100100111=>6 001100011101=>7 001100011011=>6 001100010111=>5 001100001111=>4 001011010101=>4 001011010011=>3 001011001101=>3 001011001011=>2 001011000111=>1 001010110101=>8 001010110011=>7 001010101101=>7 001010101011=>6 001010100111=>5 001010011101=>6 001010011011=>5 001010010111=>4 001010001111=>3 001001110101=>7 001001110011=>6 001001101101=>6 001001101011=>5 001001100111=>4 001001011101=>5 001001011011=>4 001001010111=>3 001001001111=>2 001000111101=>7 001000111011=>6 001000110111=>5 001000101111=>4 001000011111=>3 000111010101=>3 000111010011=>2 000111001101=>2 000111001011=>1 000111000111=>0 000110110101=>7 000110110011=>6 000110101101=>6 000110101011=>5 000110100111=>4 000110011101=>5 000110011011=>4 000110010111=>3 000110001111=>2 000101110101=>6 000101110011=>5 000101101101=>5 000101101011=>4 000101100111=>3 000101011101=>4 000101011011=>3 000101010111=>2 000101001111=>1 000100111101=>6 000100111011=>5 000100110111=>4 000100101111=>3 000100011111=>2 000011110101=>5 000011110011=>4 000011101101=>4 000011101011=>3 000011100111=>2 000011011101=>3 000011011011=>2 000011010111=>1 000011001111=>0 000010111101=>5 000010111011=>4 000010110111=>3 000010101111=>2 000010011111=>1 000001111101=>4 000001111011=>3 000001110111=>2 000001101111=>1 000001011111=>0 000000111111=>0 111110000001=>5 111001100001=>10 111000011001=>13 111000110001=>15 111000100101=>13 111011000001=>3 111010010001=>10 111010000101=>8 100111100001=>6 100110011001=>9 100110110001=>11 100110100101=>9 101101100001=>8 101100011001=>11 101100110001=>13 101100100101=>11 101111000001=>1 101110010001=>8 101110000101=>6 101001011001=>9 101001110001=>11 101001100101=>9 101011010001=>6 101011000101=>4 101010010101=>9 000001111110=>5 000111100110=>5 000111111000=>9 000111110010=>7 000111001110=>3 000111011010=>5 000101101110=>6 000101111010=>8 011001100110=>9 011001111000=>13 011001110010=>11 011001011010=>9 011110000110=>6 011110011000=>10 011110010010=>8 011111100000=>5 011111001000=>3 011111000010=>1 011100100110=>11 011100111000=>15 011100110010=>13 011100011010=>11 011101101000=>10 011101100010=>8 011101001010=>6 010011111000=>11 010011110010=>9 010011011010=>7 010001111010=>10 010110100110=>9 010110111000=>13 010110110010=>11 010110011010=>9 010111101000=>8 010111100010=>6 010111001010=>4 010101101010=>9 010101010110=>7 010101011001=>7 010101011010=>8 010101011100=>9 010101100101=>7 010101100110=>8 010101101001=>8 010101101100=>10 010101110001=>9 010101110010=>10 010101110100=>11 010101111000=>12 010110010101=>7 010110010110=>8 010110011001=>8 010110011100=>10 010110100101=>8 010110101001=>9 010110101010=>10 010110101100=>11 010110110001=>10 010110110100=>12 010111000101=>2 010111000110=>3 010111001001=>3 010111001100=>5 010111010001=>4 010111010010=>5 010111010100=>6 010111011000=>7 010111100001=>5 010111100100=>7 010111110000=>9 011001010101=>7 011001010110=>8 011001011001=>8 011001011100=>10 011001100101=>8 011001101001=>9 011001101010=>10 011001101100=>11 011001110001=>10 011001110100=>12 011010010101=>8 011010010110=>9 011010011001=>9 011010011010=>10 011010011100=>11 011010100101=>9 011010100110=>10 011010101001=>10 011010101010=>11 011010101100=>12 011010110001=>11 011010110010=>12 011010110100=>13 011010111000=>14 011011000101=>3 011011000110=>4 011011001001=>4 011011001010=>5 011011001100=>6 011011010001=>5 011011010010=>6 011011010100=>7 011011011000=>8 011011100001=>6 011011100010=>7 011011100100=>8 011011101000=>9 011011110000=>10 011100010101=>9 011100010110=>10 011100011001=>10 011100011100=>12 011100100101=>10 011100101001=>11 011100101010=>12 011100101100=>13 011100110001=>12 011100110100=>14 011101000101=>4 011101000110=>5 011101001001=>5 011101001100=>7 011101010001=>6 011101010010=>7 011101010100=>8 011101011000=>9 011101100001=>7 011101100100=>9 011101110000=>11 011110000101=>5 011110001001=>6 011110001010=>7 011110001100=>8 011110010001=>7 011110010100=>9 011110100001=>8 011110100010=>9 011110100100=>10 011110101000=>11 011110110000=>12 011111000001=>0 011111000100=>2 011111010000=>4 001111000110=>2 001111001010=>3 100111001100=>6 001111001100=>4 001011010110=>5 001111010010=>4 100111010100=>7 001111010100=>5 000111010110=>4 001011011010=>6 100111011000=>8 001111011000=>6 100011011100=>9 010011011100=>8 001011011100=>7 000111011100=>6 000011011110=>4 001101100110=>7 001101101010=>8 100101101100=>11 001101101100=>9 001011100110=>6 001111100010=>5 100111100100=>8 001111100100=>6 010011101010=>8 001011101010=>7 100111101000=>9 001111101000=>7 000111101010=>6 100011101100=>10 010011101100=>9 001011101100=>8 000111101100=>7 000011101110=>5 101001110100=>13 001001110110=>8 001101110010=>9 100101110100=>12 001101110100=>10 000101110110=>7 001011110010=>8 100111110000=>10 001111110000=>8 100011110100=>11 010011110100=>10 001011110100=>9 000111110100=>8 000011110110=>6 110001111000=>15 101001111000=>14 001001111010=>9 100101111000=>13 001101111000=>11 100011111000=>12 001011111000=>10 000011111010=>7 010001111100=>11 001001111100=>10 000101111100=>9 000011111100=>8 100101010101=>7 100101011001=>8 100101100101=>8 100101101001=>9 100101110001=>10 100110010101=>8 100110101001=>10 100111000101=>3 100111001001=>4 100111010001=>5 101001010101=>8 101001101001=>10 101010011001=>10 101010100101=>10 101010101001=>11 101010110001=>12 101011001001=>5 101011100001=>7 101100010101=>10 101100101001=>12 101101000101=>5 101101001001=>6 101101010001=>7 101110001001=>7 101110100001=>9 110001010101=>9 110001011001=>10 110001100101=>10 110001101001=>11 110001110001=>12 110010010101=>10 110010011001=>11 110010100101=>11 110010101001=>12 110010110001=>13 110011000101=>5 110011001001=>6 110011010001=>7 110011100001=>8 110100010101=>11 110100011001=>12 110100100101=>12 110100101001=>13 110100110001=>14 110101000101=>6 110101001001=>7 110101010001=>8 110101100001=>9 110110000101=>7 110110001001=>8 110110010001=>9 110110100001=>10 110111000001=>2 111000010101=>12 111000101001=>14 111001000101=>7 111001001001=>8 111001010001=>9 111010001001=>9 111010100001=>11 111100000101=>9 111100001001=>10 111100010001=>11 111100100001=>12 111101000001=>4 10110111100000=>14 10111011010000=>14 10111011100000=>15 10111100110000=>14 10111101001000=>14 10111101010000=>15 10111101100000=>16 10111110000100=>3 10111110001000=>4 10111110010000=>5 10111110100000=>6 10111111000000=>7 11001111100000=>14 11011110000010=>3 11101101000010=>14 11101110000010=>4 11110011000010=>14 11110100100010=>14 11110101000010=>15 11110110000010=>5 11111000001100=>14 11111000010010=>14 11111000100010=>15 11111001000010=>16 11111010000010=>6 11111100000010=>7 1011110111000000=>18 1011111010100000=>18 1011111011000000=>19 1011111100010000=>5 1011111100100000=>6 1011111101000000=>7 1011111110000000=>8 1111011100000010=>5 1111101010000010=>18 1111101100000010=>6 1111110001000010=>18 1111110010000010=>19 1111110100000010=>7 1111111000000010=>8 111111110000000010=>9 101111111100000000=>9 111111101000000010=>8 101111111010000000=>8 111111100100000010=>22 111111011000000010=>7 101111111001000000=>7 101111110110000000=>22 11111111100000000010=>10 10111111111000000000=>10 11111111010000000010=>9 10111111110100000000=>9 1111111111000000000010=>11 1011111111110000000000=>11
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Description
The number of internal inversions of a binary word.
Let $\bar w$ be the non-decreasing rearrangement of $w$, that is, $\bar w$ is sorted.
An internal inversion is a pair $i < j$ such that $w_i > w_j$ and $\bar w_i = \bar w_j$. For example, the word $110$ has two inversions, but only the second is internal.
References
[1] Foata, D., Han, G.-N. Inverses of words MathSciNet:1611988
[2] Hohlweg, C., Reutenauer, C. Inverses of words and the parabolic structure of the symmetric group MathSciNet:1861050
Code
def statistic(w):
    w_bar = sorted(w)
    return len([(i,j) for (i, j) in w.inversions() if w_bar[i] == w_bar[j]])
Created
Aug 01, 2017 at 08:16 by Martin Rubey
Updated
Apr 19, 2019 at 22:18 by Martin Rubey