edit this statistic or download as text // json
Identifier
Values
=>
0=>0 1=>0 00=>0 01=>0 10=>1 11=>0 000=>0 001=>0 010=>1 011=>0 100=>1 101=>1 110=>1 111=>0 0000=>0 0001=>0 0010=>1 0011=>0 0100=>1 0101=>1 0110=>1 0111=>0 1000=>1 1001=>1 1010=>2 1011=>1 1100=>2 1101=>1 1110=>1 1111=>0 00000=>0 00001=>0 00010=>1 00011=>0 00100=>1 00101=>1 00110=>1 00111=>0 01000=>1 01001=>1 01010=>2 01011=>1 01100=>2 01101=>1 01110=>1 01111=>0 10000=>1 10001=>1 10010=>1 10011=>1 10100=>2 10101=>2 10110=>1 10111=>1 11000=>2 11001=>2 11010=>2 11011=>1 11100=>2 11101=>1 11110=>1 11111=>0 000000=>0 000001=>0 000010=>1 000011=>0 000100=>1 000101=>1 000110=>1 000111=>0 001000=>1 001001=>1 001010=>2 001011=>1 001100=>2 001101=>1 001110=>1 001111=>0 010000=>1 010001=>1 010010=>1 010011=>1 010100=>2 010101=>2 010110=>1 010111=>1 011000=>2 011001=>2 011010=>2 011011=>1 011100=>2 011101=>1 011110=>1 011111=>0 100000=>1 100001=>1 100010=>1 100011=>1 100100=>1 100101=>1 100110=>1 100111=>1 101000=>2 101001=>2 101010=>3 101011=>2 101100=>3 101101=>1 101110=>1 101111=>1 110000=>2 110001=>2 110010=>3 110011=>2 110100=>3 110101=>2 110110=>1 110111=>1 111000=>3 111001=>2 111010=>2 111011=>1 111100=>2 111101=>1 111110=>1 111111=>0 0000000=>0 0000001=>0 0000010=>1 0000011=>0 0000100=>1 0000101=>1 0000110=>1 0000111=>0 0001000=>1 0001001=>1 0001010=>2 0001011=>1 0001100=>2 0001101=>1 0001110=>1 0001111=>0 0010000=>1 0010001=>1 0010010=>1 0010011=>1 0010100=>2 0010101=>2 0010110=>1 0010111=>1 0011000=>2 0011001=>2 0011010=>2 0011011=>1 0011100=>2 0011101=>1 0011110=>1 0011111=>0 0100000=>1 0100001=>1 0100010=>1 0100011=>1 0100100=>1 0100101=>1 0100110=>1 0100111=>1 0101000=>2 0101001=>2 0101010=>3 0101011=>2 0101100=>3 0101101=>1 0101110=>1 0101111=>1 0110000=>2 0110001=>2 0110010=>3 0110011=>2 0110100=>3 0110101=>2 0110110=>1 0110111=>1 0111000=>3 0111001=>2 0111010=>2 0111011=>1 0111100=>2 0111101=>1 0111110=>1 0111111=>0 1000000=>1 1000001=>1 1000010=>1 1000011=>1 1000100=>1 1000101=>1 1000110=>1 1000111=>1 1001000=>1 1001001=>1 1001010=>2 1001011=>1 1001100=>2 1001101=>1 1001110=>1 1001111=>1 1010000=>2 1010001=>2 1010010=>2 1010011=>2 1010100=>3 1010101=>3 1010110=>2 1010111=>2 1011000=>3 1011001=>3 1011010=>2 1011011=>1 1011100=>2 1011101=>1 1011110=>1 1011111=>1 1100000=>2 1100001=>2 1100010=>2 1100011=>2 1100100=>3 1100101=>3 1100110=>2 1100111=>2 1101000=>3 1101001=>3 1101010=>3 1101011=>2 1101100=>3 1101101=>1 1101110=>1 1101111=>1 1110000=>3 1110001=>3 1110010=>3 1110011=>2 1110100=>3 1110101=>2 1110110=>1 1110111=>1 1111000=>3 1111001=>2 1111010=>2 1111011=>1 1111100=>2 1111101=>1 1111110=>1 1111111=>0 00000000=>0 00000001=>0 00000010=>1 00000011=>0 00000100=>1 00000101=>1 00000110=>1 00000111=>0 00001000=>1 00001001=>1 00001010=>2 00001011=>1 00001100=>2 00001101=>1 00001110=>1 00001111=>0 00010000=>1 00010001=>1 00010010=>1 00010011=>1 00010100=>2 00010101=>2 00010110=>1 00010111=>1 00011000=>2 00011001=>2 00011010=>2 00011011=>1 00011100=>2 00011101=>1 00011110=>1 00011111=>0 00100000=>1 00100001=>1 00100010=>1 00100011=>1 00100100=>1 00100101=>1 00100110=>1 00100111=>1 00101000=>2 00101001=>2 00101010=>3 00101011=>2 00101100=>3 00101101=>1 00101110=>1 00101111=>1 00110000=>2 00110001=>2 00110010=>3 00110011=>2 00110100=>3 00110101=>2 00110110=>1 00110111=>1 00111000=>3 00111001=>2 00111010=>2 00111011=>1 00111100=>2 00111101=>1 00111110=>1 00111111=>0 01000000=>1 01000001=>1 01000010=>1 01000011=>1 01000100=>1 01000101=>1 01000110=>1 01000111=>1 01001000=>1 01001001=>1 01001010=>2 01001011=>1 01001100=>2 01001101=>1 01001110=>1 01001111=>1 01010000=>2 01010001=>2 01010010=>2 01010011=>2 01010100=>3 01010101=>3 01010110=>2 01010111=>2 01011000=>3 01011001=>3 01011010=>2 01011011=>1 01011100=>2 01011101=>1 01011110=>1 01011111=>1 01100000=>2 01100001=>2 01100010=>2 01100011=>2 01100100=>3 01100101=>3 01100110=>2 01100111=>2 01101000=>3 01101001=>3 01101010=>3 01101011=>2 01101100=>3 01101101=>1 01101110=>1 01101111=>1 01110000=>3 01110001=>3 01110010=>3 01110011=>2 01110100=>3 01110101=>2 01110110=>1 01110111=>1 01111000=>3 01111001=>2 01111010=>2 01111011=>1 01111100=>2 01111101=>1 01111110=>1 01111111=>0 10000000=>1 10000001=>1 10000010=>1 10000011=>1 10000100=>1 10000101=>1 10000110=>1 10000111=>1 10001000=>1 10001001=>1 10001010=>2 10001011=>1 10001100=>2 10001101=>1 10001110=>1 10001111=>1 10010000=>1 10010001=>1 10010010=>1 10010011=>1 10010100=>2 10010101=>2 10010110=>1 10010111=>1 10011000=>2 10011001=>2 10011010=>2 10011011=>1 10011100=>2 10011101=>1 10011110=>1 10011111=>1 10100000=>2 10100001=>2 10100010=>2 10100011=>2 10100100=>2 10100101=>2 10100110=>2 10100111=>2 10101000=>3 10101001=>3 10101010=>4 10101011=>3 10101100=>4 10101101=>2 10101110=>2 10101111=>2 10110000=>3 10110001=>3 10110010=>4 10110011=>3 10110100=>4 10110101=>2 10110110=>1 10110111=>1 10111000=>4 10111001=>2 10111010=>2 10111011=>1 10111100=>2 10111101=>1 10111110=>1 10111111=>1 11000000=>2 11000001=>2 11000010=>2 11000011=>2 11000100=>2 11000101=>2 11000110=>2 11000111=>2 11001000=>3 11001001=>3 11001010=>4 11001011=>3 11001100=>4 11001101=>2 11001110=>2 11001111=>2 11010000=>3 11010001=>3 11010010=>4 11010011=>3 11010100=>4 11010101=>3 11010110=>2 11010111=>2 11011000=>4 11011001=>3 11011010=>2 11011011=>1 11011100=>2 11011101=>1 11011110=>1 11011111=>1 11100000=>3 11100001=>3 11100010=>4 11100011=>3 11100100=>4 11100101=>3 11100110=>2 11100111=>2 11101000=>4 11101001=>3 11101010=>3 11101011=>2 11101100=>3 11101101=>1 11101110=>1 11101111=>1 11110000=>4 11110001=>3 11110010=>3 11110011=>2 11110100=>3 11110101=>2 11110110=>1 11110111=>1 11111000=>3 11111001=>2 11111010=>2 11111011=>1 11111100=>2 11111101=>1 11111110=>1 11111111=>0 000000000=>0 000000001=>0 000000010=>1 000000011=>0 000000100=>1 000000101=>1 000000110=>1 000000111=>0 000001000=>1 000001001=>1 000001010=>2 000001011=>1 000001100=>2 000001101=>1 000001110=>1 000001111=>0 000010000=>1 000010001=>1 000010010=>1 000010011=>1 000010100=>2 000010101=>2 000010110=>1 000010111=>1 000011000=>2 000011001=>2 000011010=>2 000011011=>1 000011100=>2 000011101=>1 000011110=>1 000011111=>0 000100000=>1 000100001=>1 000100010=>1 000100011=>1 000100100=>1 000100101=>1 000100110=>1 000100111=>1 000101000=>2 000101001=>2 000101010=>3 000101011=>2 000101100=>3 000101101=>1 000101110=>1 000101111=>1 000110000=>2 000110001=>2 000110010=>3 000110011=>2 000110100=>3 000110101=>2 000110110=>1 000110111=>1 000111000=>3 000111001=>2 000111010=>2 000111011=>1 000111100=>2 000111101=>1 000111110=>1 000111111=>0 001000000=>1 001000001=>1 001000010=>1 001000011=>1 001000100=>1 001000101=>1 001000110=>1 001000111=>1 001001000=>1 001001001=>1 001001010=>2 001001011=>1 001001100=>2 001001101=>1 001001110=>1 001001111=>1 001010000=>2 001010001=>2 001010010=>2 001010011=>2 001010100=>3 001010101=>3 001010110=>2 001010111=>2 001011000=>3 001011001=>3 001011010=>2 001011011=>1 001011100=>2 001011101=>1 001011110=>1 001011111=>1 001100000=>2 001100001=>2 001100010=>2 001100011=>2 001100100=>3 001100101=>3 001100110=>2 001100111=>2 001101000=>3 001101001=>3 001101010=>3 001101011=>2 001101100=>3 001101101=>1 001101110=>1 001101111=>1 001110000=>3 001110001=>3 001110010=>3 001110011=>2 001110100=>3 001110101=>2 001110110=>1 001110111=>1 001111000=>3 001111001=>2 001111010=>2 001111011=>1 001111100=>2 001111101=>1 001111110=>1 001111111=>0 010000000=>1 010000001=>1 010000010=>1 010000011=>1 010000100=>1 010000101=>1 010000110=>1 010000111=>1 010001000=>1 010001001=>1 010001010=>2 010001011=>1 010001100=>2 010001101=>1 010001110=>1 010001111=>1 010010000=>1 010010001=>1 010010010=>1 010010011=>1 010010100=>2 010010101=>2 010010110=>1 010010111=>1 010011000=>2 010011001=>2 010011010=>2 010011011=>1 010011100=>2 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=>4 010101101=>2 010101110=>2 010101111=>2 010110000=>3 010110001=>3 010110010=>4 010110011=>3 010110100=>4 010110101=>2 010110110=>1 010110111=>1 010111000=>4 010111001=>2 010111010=>2 010111011=>1 010111100=>2 010111101=>1 010111110=>1 010111111=>1 011000000=>2 011000001=>2 011000010=>2 011000011=>2 011000100=>2 011000101=>2 011000110=>2 011000111=>2 011001000=>3 011001001=>3 011001010=>4 011001011=>3 011001100=>4 011001101=>2 011001110=>2 011001111=>2 011010000=>3 011010001=>3 011010010=>4 011010011=>3 011010100=>4 011010101=>3 011010110=>2 011010111=>2 011011000=>4 011011001=>3 011011010=>2 011011011=>1 011011100=>2 011011101=>1 011011110=>1 011011111=>1 011100000=>3 011100001=>3 011100010=>4 011100011=>3 011100100=>4 011100101=>3 011100110=>2 011100111=>2 011101000=>4 011101001=>3 011101010=>3 011101011=>2 011101100=>3 011101101=>1 011101110=>1 011101111=>1 011110000=>4 011110001=>3 011110010=>3 011110011=>2 011110100=>3 011110101=>2 011110110=>1 011110111=>1 011111000=>3 011111001=>2 011111010=>2 011111011=>1 011111100=>2 011111101=>1 011111110=>1 011111111=>0 100000000=>1 100000001=>1 100000010=>1 100000011=>1 100000100=>1 100000101=>1 100000110=>1 100000111=>1 100001000=>1 100001001=>1 100001010=>2 100001011=>1 100001100=>2 100001101=>1 100001110=>1 100001111=>1 100010000=>1 100010001=>1 100010010=>1 100010011=>1 100010100=>2 100010101=>2 100010110=>1 100010111=>1 100011000=>2 100011001=>2 100011010=>2 100011011=>1 100011100=>2 100011101=>1 100011110=>1 100011111=>1 100100000=>1 100100001=>1 100100010=>1 100100011=>1 100100100=>1 100100101=>1 100100110=>1 100100111=>1 100101000=>2 100101001=>2 100101010=>3 100101011=>2 100101100=>3 100101101=>1 100101110=>1 100101111=>1 100110000=>2 100110001=>2 100110010=>3 100110011=>2 100110100=>3 100110101=>2 100110110=>1 100110111=>1 100111000=>3 100111001=>2 100111010=>2 100111011=>1 100111100=>2 100111101=>1 100111110=>1 100111111=>1 101000000=>2 101000001=>2 101000010=>2 101000011=>2 101000100=>2 101000101=>2 101000110=>2 101000111=>2 101001000=>2 101001001=>2 101001010=>2 101001011=>2 101001100=>2 101001101=>2 101001110=>2 101001111=>2 101010000=>3 101010001=>3 101010010=>3 101010011=>3 101010100=>4 101010101=>4 101010110=>3 101010111=>3 101011000=>4 101011001=>4 101011010=>2 101011011=>2 101011100=>2 101011101=>2 101011110=>2 101011111=>2 101100000=>3 101100001=>3 101100010=>3 101100011=>3 101100100=>4 101100101=>4 101100110=>3 101100111=>3 101101000=>4 101101001=>4 101101010=>3 101101011=>2 101101100=>3 101101101=>1 101101110=>1 101101111=>1 101110000=>4 101110001=>4 101110010=>3 101110011=>2 101110100=>3 101110101=>2 101110110=>1 101110111=>1 101111000=>3 101111001=>2 101111010=>2 101111011=>1 101111100=>2 101111101=>1 101111110=>1 101111111=>1 110000000=>2 110000001=>2 110000010=>2 110000011=>2 110000100=>2 110000101=>2 110000110=>2 110000111=>2 110001000=>2 110001001=>2 110001010=>2 110001011=>2 110001100=>2 110001101=>2 110001110=>2 110001111=>2 110010000=>3 110010001=>3 110010010=>3 110010011=>3 110010100=>4 110010101=>4 110010110=>3 110010111=>3 110011000=>4 110011001=>4 110011010=>2 110011011=>2 110011100=>2 110011101=>2 110011110=>2 110011111=>2 110100000=>3 110100001=>3 110100010=>3 110100011=>3 110100100=>4 110100101=>4 110100110=>3 110100111=>3 110101000=>4 110101001=>4 110101010=>4 110101011=>3 110101100=>4 110101101=>2 110101110=>2 110101111=>2 110110000=>4 110110001=>4 110110010=>4 110110011=>3 110110100=>4 110110101=>2 110110110=>1 110110111=>1 110111000=>4 110111001=>2 110111010=>2 110111011=>1 110111100=>2 110111101=>1 110111110=>1 110111111=>1 111000000=>3 111000001=>3 111000010=>3 111000011=>3 111000100=>4 111000101=>4 111000110=>3 111000111=>3 111001000=>4 111001001=>4 111001010=>4 111001011=>3 111001100=>4 111001101=>2 111001110=>2 111001111=>2 111010000=>4 111010001=>4 111010010=>4 111010011=>3 111010100=>4 111010101=>3 111010110=>2 111010111=>2 111011000=>4 111011001=>3 111011010=>2 111011011=>1 111011100=>2 111011101=>1 111011110=>1 111011111=>1 111100000=>4 111100001=>4 111100010=>4 111100011=>3 111100100=>4 111100101=>3 111100110=>2 111100111=>2 111101000=>4 111101001=>3 111101010=>3 111101011=>2 111101100=>3 111101101=>1 111101110=>1 111101111=>1 111110000=>4 111110001=>3 111110010=>3 111110011=>2 111110100=>3 111110101=>2 111110110=>1 111110111=>1 111111000=>3 111111001=>2 111111010=>2 111111011=>1 111111100=>2 111111101=>1 111111110=>1 111111111=>0 1010101010=>5 1010101100=>5 1010110010=>5 1010110100=>5 1010111000=>5 1011001010=>5 1011001100=>5 1011010010=>5 1011010100=>5 1011011000=>5 1011100010=>5 1011100100=>5 1011101000=>5 1011110000=>5 1100101010=>5 1100101100=>5 1100110010=>5 1100110100=>5 1100111000=>5 1101001010=>5 1101001100=>5 1101010010=>5 1101010100=>5 1101011000=>5 1101100010=>5 1101100100=>5 1101101000=>5 1101110000=>5 1110001010=>5 1110001100=>5 1110010010=>5 1110010100=>5 1110011000=>5 1110100010=>5 1110100100=>5 1110101000=>5 1110110000=>5 1111000010=>5 1111000100=>5 1111001000=>5 1111010000=>5 1111100000=>5 1111111110=>1 1111111101=>1 1111000001=>4 1110111110=>1 1110100001=>4 1110011110=>2 1110010001=>4 1110001001=>4 1110000101=>3 1101111110=>1 1101100001=>4 1101010001=>4 1101001110=>3 1101001001=>4 1101000101=>3 1100110001=>4 1100101001=>4 1100100110=>3 1100100101=>3 1100011001=>2 1100010101=>2 1100010010=>2 1100001000=>2 1011111110=>1 1011111101=>1 1011100001=>4 1011010001=>4 1011001001=>4 1011000101=>3 1010111110=>2 1010110001=>4 1010101001=>4 1010100110=>3 1010100101=>3 1010011010=>2 1010011001=>2 1010010110=>2 1010010101=>2 1010001110=>2 1001111110=>1 1001111000=>3 1001110100=>3 1001110001=>3 1001101100=>3 1001101010=>3 1001101001=>3 1001100110=>2 1001100101=>3 1001011110=>1 1001011010=>2 1001011001=>3 1001010110=>2 1001010101=>3 1001001110=>1 1000111110=>1 1000111100=>2 1000111010=>2 1000110110=>1 1000101110=>1 1000011110=>1 1000010110=>1 1000001110=>1 1000001010=>2 1000000110=>1 1000000100=>1 1000000010=>1 1000000000=>1 0111111111=>0 0111111101=>1 0111111001=>2 0111110001=>3 0111100001=>4 0111000001=>3 0110000001=>2 0100000001=>1 0000000001=>0 0111111011=>1 0111110101=>2 0111101001=>3 0111010001=>4 0110100001=>3 0101000001=>2 0010000001=>1 0101010101=>4 0101010011=>3 0101001101=>2 0101001011=>2 0101000111=>2 0100110101=>2 0100110011=>2 0100101101=>1 0100101011=>2 0100100111=>1 0100011101=>1 0100011011=>1 0100010111=>1 0100001111=>1 0011010101=>3 0011010011=>3 0011001101=>2 0011001011=>3 0011000111=>2 0010110101=>2 0010110011=>3 0010101101=>2 0010101011=>3 0010100111=>2 0010011101=>1 0010011011=>1 0010010111=>1 0010001111=>1 0001110101=>2 0001110011=>2 0001101101=>1 0001101011=>2 0001100111=>2 0001011101=>1 0001011011=>1 0001010111=>2 0001001111=>1 0000111101=>1 0000111011=>1 0000110111=>1 0000101111=>1 0000011111=>0 0000000000=>0 0000000110=>1 0000011000=>2 0000011110=>1 0000010010=>1 0001100000=>2 0001100110=>2 0001111000=>3 0001111110=>1 0001110010=>3 0001001000=>1 0001001110=>1 0001000010=>1 0001011010=>2 0100000010=>1 0010101010=>4 0010101000=>3 0010101100=>4 0010100100=>2 0010010100=>2 0001010100=>3 0010101110=>2 0010100000=>2 0010111100=>2 0010000100=>1 0011110100=>3 0000010100=>2 0111010100=>4 0000000010=>1 0000001110=>1 0000111110=>1 0000000100=>1 0000001010=>2 0000010110=>1 0000101110=>1 0001011110=>1 0100000000=>1 0100000011=>1 0100000111=>1 0100011111=>1 0100111111=>1 0101111111=>1 0100000101=>1 0100001011=>1 0100101111=>1 0101011111=>2 0110111111=>1 0101010110=>3 0101011001=>4 0101011010=>2 0101011100=>2 0101100101=>4 0101100110=>3 0101101001=>4 0101101010=>3 0101101100=>3 0101110001=>4 0101110010=>3 0101110100=>3 0101111000=>3 0110010101=>4 0110010110=>3 0110011001=>4 0110011010=>2 0110011100=>2 0110100101=>4 0110100110=>3 0110101001=>4 0110101010=>4 0110101100=>4 0110110001=>4 0110110010=>4 0110110100=>4 0110111000=>4 0111000101=>4 0111000110=>3 0111001001=>4 0111001010=>4 0111001100=>4 0111010010=>4 0111011000=>4 0111100010=>4 0111100100=>4 0111101000=>4 0111110000=>4 0100000100=>1 0011100110=>2 0011101010=>3 0011101100=>3 0001101110=>1 0010110110=>1 0011110010=>3 0001110110=>1 0100111010=>2 0010111010=>2 0011111000=>3 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110010110100=>6 110010111000=>6 110011001010=>6 110011001100=>6 110011010010=>6 110011010100=>6 110011011000=>6 110011100010=>6 110011100100=>6 110011101000=>6 110011110000=>6 110100101010=>6 110100101100=>6 110100110010=>6 110100110100=>6 110100111000=>6 110101001010=>6 110101001100=>6 110101010010=>6 110101010100=>6 110101011000=>6 110101100010=>6 110101100100=>6 110101101000=>6 110101110000=>6 110110001010=>6 110110001100=>6 110110010010=>6 110110010100=>6 110110011000=>6 110110100010=>6 110110100100=>6 110110101000=>6 110110110000=>6 110111000010=>6 110111000100=>6 110111001000=>6 110111010000=>6 110111100000=>6 111000101010=>6 111000101100=>6 111000110010=>6 111000110100=>6 111000111000=>6 111001001010=>6 111001001100=>6 111001010010=>6 111001010100=>6 111001011000=>6 111001100010=>6 111001100100=>6 111001101000=>6 111001110000=>6 111010001010=>6 111010001100=>6 111010010010=>6 111010010100=>6 111010011000=>6 111010100010=>6 111010100100=>6 111010101000=>6 111010110000=>6 111011000010=>6 111011000100=>6 111011001000=>6 111011010000=>6 111011100000=>6 111100001010=>6 111100001100=>6 111100010010=>6 111100010100=>6 111100011000=>6 111100100010=>6 111100100100=>6 111100101000=>6 111100110000=>6 111101000010=>6 111101000100=>6 111101001000=>6 111101010000=>6 111101100000=>6 111110000010=>6 111110000100=>6 111110001000=>6 111110010000=>6 111110100000=>6 111111000000=>6 100001111100=>2 010101010101=>5 010101010011=>4 010101001101=>3 010101001011=>3 010101000111=>3 010100110101=>2 010100110011=>2 010100101101=>2 010100101011=>2 010100100111=>2 010100011101=>2 010100011011=>2 010100010111=>2 010100001111=>2 010011010101=>3 010011010011=>3 010011001101=>2 010011001011=>3 010011000111=>2 010010110101=>2 010010110011=>3 010010101101=>2 010010101011=>3 010010100111=>2 010010011101=>1 010010011011=>1 010010010111=>1 010010001111=>1 010001110101=>2 010001110011=>2 010001101101=>1 010001101011=>2 010001100111=>2 010001011101=>1 010001011011=>1 010001010111=>2 010001001111=>1 010000111101=>1 010000111011=>1 010000110111=>1 010000101111=>1 010000011111=>1 001101010101=>4 001101010011=>4 001101001101=>3 001101001011=>4 001101000111=>3 001100110101=>2 001100110011=>4 001100101101=>3 001100101011=>4 001100100111=>3 001100011101=>2 001100011011=>2 001100010111=>2 001100001111=>2 001011010101=>3 001011010011=>4 001011001101=>3 001011001011=>4 001011000111=>3 001010110101=>2 001010110011=>4 001010101101=>3 001010101011=>4 001010100111=>3 001010011101=>2 001010011011=>2 001010010111=>2 001010001111=>2 001001110101=>2 001001110011=>2 001001101101=>1 001001101011=>2 001001100111=>2 001001011101=>1 001001011011=>1 001001010111=>2 001001001111=>1 001000111101=>1 001000111011=>1 001000110111=>1 001000101111=>1 001000011111=>1 000111010101=>3 000111010011=>3 000111001101=>2 000111001011=>3 000111000111=>3 000110110101=>2 000110110011=>3 000110101101=>2 000110101011=>3 000110100111=>3 000110011101=>2 000110011011=>2 000110010111=>3 000110001111=>2 000101110101=>2 000101110011=>2 000101101101=>1 000101101011=>2 000101100111=>3 000101011101=>2 000101011011=>2 000101010111=>3 000101001111=>2 000100111101=>1 000100111011=>1 000100110111=>1 000100101111=>1 000100011111=>1 000011110101=>2 000011110011=>2 000011101101=>1 000011101011=>2 000011100111=>2 000011011101=>1 000011011011=>1 000011010111=>2 000011001111=>2 000010111101=>1 000010111011=>1 000010110111=>1 000010101111=>2 000010011111=>1 000001111101=>1 000001111011=>1 000001110111=>1 000001101111=>1 000001011111=>1 000000111111=>0 111110000001=>5 111001100001=>5 111000011001=>3 111000110001=>5 111000100101=>4 111011000001=>5 111010010001=>5 111010000101=>4 100111100001=>4 100110011001=>4 100110110001=>4 100110100101=>4 101101100001=>5 101100011001=>3 101100110001=>5 101100100101=>4 101111000001=>5 101110010001=>5 101110000101=>4 101001011001=>3 101001110001=>3 101001100101=>3 101011010001=>5 101011000101=>4 101010010101=>3 000001111110=>1 000111100110=>2 000111111000=>3 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010111010100=>4 010111011000=>4 010111100001=>5 010111100100=>4 010111110000=>4 011001010101=>5 011001010110=>4 011001011001=>5 011001011100=>3 011001100101=>5 011001101001=>5 011001101010=>3 011001101100=>3 011001110001=>5 011001110100=>3 011010010101=>5 011010010110=>4 011010011001=>5 011010011010=>3 011010011100=>3 011010100101=>5 011010100110=>4 011010101001=>5 011010101010=>5 011010101100=>5 011010110001=>5 011010110010=>5 011010110100=>5 011010111000=>5 011011000101=>5 011011000110=>4 011011001001=>5 011011001010=>5 011011001100=>5 011011010001=>5 011011010010=>5 011011010100=>5 011011011000=>5 011011100001=>5 011011100010=>5 011011100100=>5 011011101000=>5 011011110000=>5 011100010101=>5 011100010110=>4 011100011001=>5 011100011100=>3 011100100101=>5 011100101001=>5 011100101010=>5 011100101100=>5 011100110001=>5 011100110100=>5 011101000101=>5 011101000110=>4 011101001001=>5 011101001100=>5 011101010001=>5 011101010010=>5 011101010100=>5 011101011000=>5 011101100001=>5 011101100100=>5 011101110000=>5 011110000101=>5 011110001001=>5 011110001010=>5 011110001100=>5 011110010001=>5 011110010100=>5 011110100001=>5 011110100010=>5 011110100100=>5 011110101000=>5 011110110000=>5 011111000001=>5 011111000100=>5 011111010000=>5 001111000110=>3 001111001010=>4 100111001100=>4 001111001100=>4 001011010110=>2 001111010010=>4 100111010100=>4 001111010100=>4 000111010110=>2 001011011010=>2 100111011000=>4 001111011000=>4 100011011100=>2 010011011100=>2 001011011100=>2 000111011100=>2 000011011110=>1 001101100110=>3 001101101010=>3 100101101100=>3 001101101100=>3 001011100110=>2 001111100010=>4 100111100100=>4 001111100100=>4 010011101010=>3 001011101010=>3 100111101000=>4 001111101000=>4 000111101010=>3 100011101100=>3 010011101100=>3 001011101100=>3 000111101100=>3 000011101110=>1 101001110100=>3 001001110110=>1 001101110010=>3 100101110100=>3 001101110100=>3 000101110110=>1 001011110010=>3 100111110000=>4 001111110000=>4 100011110100=>3 010011110100=>3 001011110100=>3 000111110100=>3 000011110110=>1 110001111000=>3 101001111000=>3 001001111010=>2 100101111000=>3 001101111000=>3 100011111000=>3 001011111000=>3 000011111010=>2 010001111100=>2 001001111100=>2 000101111100=>2 000011111100=>2 100101010101=>4 100101011001=>4 100101100101=>4 100101101001=>4 100101110001=>4 100110010101=>4 100110101001=>4 100111000101=>4 100111001001=>4 100111010001=>4 101001010101=>3 101001101001=>3 101010011001=>3 101010100101=>4 101010101001=>5 101010110001=>5 101011001001=>5 101011100001=>5 101100010101=>3 101100101001=>5 101101000101=>4 101101001001=>5 101101010001=>5 101110001001=>5 101110100001=>5 110001010101=>3 110001011001=>3 110001100101=>3 110001101001=>3 110001110001=>3 110010010101=>3 110010011001=>3 110010100101=>4 110010101001=>5 110010110001=>5 110011000101=>4 110011001001=>5 110011010001=>5 110011100001=>5 110100010101=>3 110100011001=>3 110100100101=>4 110100101001=>5 110100110001=>5 110101000101=>4 110101001001=>5 110101010001=>5 110101100001=>5 110110000101=>4 110110001001=>5 110110010001=>5 110110100001=>5 110111000001=>5 111000010101=>3 111000101001=>5 111001000101=>4 111001001001=>5 111001010001=>5 111010001001=>5 111010100001=>5 111100000101=>4 111100001001=>5 111100010001=>5 111100100001=>5 111101000001=>5 10110111100000=>7 10111011010000=>7 10111011100000=>7 10111100110000=>7 10111101001000=>7 10111101010000=>7 10111101100000=>7 10111110000100=>7 10111110001000=>7 10111110010000=>7 10111110100000=>7 10111111000000=>7 11001111100000=>7 11011110000010=>7 11101101000010=>7 11101110000010=>7 11110011000010=>7 11110100100010=>7 11110101000010=>7 11110110000010=>7 11111000001100=>7 11111000010010=>7 11111000100010=>7 11111001000010=>7 11111010000010=>7 11111100000010=>7 1011110111000000=>8 1011111010100000=>8 1011111011000000=>8 1011111100010000=>8 1011111100100000=>8 1011111101000000=>8 1011111110000000=>8 1111011100000010=>8 1111101010000010=>8 1111101100000010=>8 1111110001000010=>8 1111110010000010=>8 1111110100000010=>8 1111111000000010=>8 111111110000000010=>9 101111111100000000=>9 111111101000000010=>9 101111111010000000=>9 111111100100000010=>9 111111011000000010=>9 101111111001000000=>9 101111110110000000=>9 11111111100000000010=>10 10111111111000000000=>10 11111111010000000010=>10 10111111110100000000=>10 1111111111000000000010=>11 1011111111110000000000=>11
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Description
The semilength of the longest Dyck word in the Catalan factorisation of a binary word.
Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2].
This statistic records the semilength of the longest Dyck word in this factorisation.
References
[1] Lothaire, M. Applied combinatorics on words MathSciNet:2165687
Code
def statistic(w):
    c = catalan_factorization(w)
    return max(len(f)/2 for f in c[0] + [c[1]] + c[2])

def first_passage_decomposition(self):
    r"""
    Decompose this word in the form (D 0)* P, where D is the family
    of complete Dyck words and P is the family of Dyck prefixes.

    The result is returned as a pair.

    EXAMPLES::

        sage: first_passage_decomposition([0,1,1,0,0,1])
        ([[]], [1, 1, 0, 0, 1])

        sage: first_passage_decomposition([0,1,1,0,0,0,1])
        ([[], [1, 1, 0, 0]), [1])
    """
    down = []
    i = 0  # start of current piece
    h = 0  # current height
    s = -1 # next minimal height
    for j, e in enumerate(self):
        if e == 1:
            h += 1
        else:
            h -= 1
        if h == s:
            down.append(self[i:j])
            i = j+1
            s -= 1
    return (down, self[i:])

def catalan_factorization(self):
    r"""
    Decompose this word in the form (D 0)* D (1 D)*, where D is the
    family of complete Dyck words.

    The result is returned as a triple.

    EXAMPLES::

        sage: catalan_factorization([])
        ([], [], [])
        sage: catalan_factorization([1])
        ([], [], [[]])
        sage: catalan_factorization([1,0,1])
        ([], [1, 0], [[]])

        sage: catalan_factorization([1,0,0,1,0,1,1,1,0,1,0,0])
        ([[1, 0]], [1, 0], [[1, 1, 0, 1, 0, 0]])

        sage: catalan_factorization([0,1,1,0,0,1,1,0,1])
        ([[]], [1, 1, 0, 0], [[1, 0], []])

    """
    down, rest = first_passage_decomposition(self)
    up, middle = first_passage_decomposition([1-e for e in rest[::-1]])
    return down, [1-e for e in middle[::-1]], [[1-e for e in p[::-1]] for p in up[::-1]]
Created
Jul 03, 2017 at 08:46 by Martin Rubey
Updated
Jan 20, 2018 at 20:12 by Martin Rubey