edit this statistic or download as text // json
Identifier
Values
=>
0=>1 1=>1 00=>2 01=>2 10=>1 11=>2 000=>3 001=>3 010=>2 011=>3 100=>1 101=>2 110=>1 111=>3 0000=>4 0001=>4 0010=>3 0011=>4 0100=>2 0101=>3 0110=>2 0111=>4 1000=>2 1001=>2 1010=>1 1011=>3 1100=>1 1101=>2 1110=>2 1111=>4 00000=>5 00001=>5 00010=>4 00011=>5 00100=>3 00101=>4 00110=>3 00111=>5 01000=>3 01001=>3 01010=>2 01011=>4 01100=>2 01101=>3 01110=>3 01111=>5 10000=>3 10001=>3 10010=>2 10011=>3 10100=>1 10101=>2 10110=>2 10111=>4 11000=>1 11001=>2 11010=>1 11011=>3 11100=>1 11101=>3 11110=>3 11111=>5 000000=>6 000001=>6 000010=>5 000011=>6 000100=>4 000101=>5 000110=>4 000111=>6 001000=>4 001001=>4 001010=>3 001011=>5 001100=>3 001101=>4 001110=>4 001111=>6 010000=>4 010001=>4 010010=>3 010011=>4 010100=>2 010101=>3 010110=>3 010111=>5 011000=>2 011001=>3 011010=>2 011011=>4 011100=>2 011101=>4 011110=>4 011111=>6 100000=>4 100001=>4 100010=>3 100011=>4 100100=>2 100101=>3 100110=>2 100111=>4 101000=>2 101001=>2 101010=>1 101011=>3 101100=>1 101101=>3 101110=>3 101111=>5 110000=>2 110001=>2 110010=>1 110011=>3 110100=>1 110101=>2 110110=>2 110111=>4 111000=>1 111001=>2 111010=>2 111011=>4 111100=>2 111101=>4 111110=>4 111111=>6 0000000=>7 0000001=>7 0000010=>6 0000011=>7 0000100=>5 0000101=>6 0000110=>5 0000111=>7 0001000=>5 0001001=>5 0001010=>4 0001011=>6 0001100=>4 0001101=>5 0001110=>5 0001111=>7 0010000=>5 0010001=>5 0010010=>4 0010011=>5 0010100=>3 0010101=>4 0010110=>4 0010111=>6 0011000=>3 0011001=>4 0011010=>3 0011011=>5 0011100=>3 0011101=>5 0011110=>5 0011111=>7 0100000=>5 0100001=>5 0100010=>4 0100011=>5 0100100=>3 0100101=>4 0100110=>3 0100111=>5 0101000=>3 0101001=>3 0101010=>2 0101011=>4 0101100=>2 0101101=>4 0101110=>4 0101111=>6 0110000=>3 0110001=>3 0110010=>2 0110011=>4 0110100=>2 0110101=>3 0110110=>3 0110111=>5 0111000=>2 0111001=>3 0111010=>3 0111011=>5 0111100=>3 0111101=>5 0111110=>5 0111111=>7 1000000=>5 1000001=>5 1000010=>4 1000011=>5 1000100=>3 1000101=>4 1000110=>3 1000111=>5 1001000=>3 1001001=>3 1001010=>2 1001011=>4 1001100=>2 1001101=>3 1001110=>3 1001111=>5 1010000=>3 1010001=>3 1010010=>2 1010011=>3 1010100=>1 1010101=>2 1010110=>2 1010111=>4 1011000=>1 1011001=>2 1011010=>2 1011011=>4 1011100=>2 1011101=>4 1011110=>4 1011111=>6 1100000=>3 1100001=>3 1100010=>2 1100011=>3 1100100=>1 1100101=>2 1100110=>2 1100111=>4 1101000=>1 1101001=>2 1101010=>1 1101011=>3 1101100=>1 1101101=>3 1101110=>3 1101111=>5 1110000=>1 1110001=>2 1110010=>1 1110011=>3 1110100=>1 1110101=>3 1110110=>3 1110111=>5 1111000=>1 1111001=>3 1111010=>3 1111011=>5 1111100=>3 1111101=>5 1111110=>5 1111111=>7 00000000=>8 00000001=>8 00000010=>7 00000011=>8 00000100=>6 00000101=>7 00000110=>6 00000111=>8 00001000=>6 00001001=>6 00001010=>5 00001011=>7 00001100=>5 00001101=>6 00001110=>6 00001111=>8 00010000=>6 00010001=>6 00010010=>5 00010011=>6 00010100=>4 00010101=>5 00010110=>5 00010111=>7 00011000=>4 00011001=>5 00011010=>4 00011011=>6 00011100=>4 00011101=>6 00011110=>6 00011111=>8 00100000=>6 00100001=>6 00100010=>5 00100011=>6 00100100=>4 00100101=>5 00100110=>4 00100111=>6 00101000=>4 00101001=>4 00101010=>3 00101011=>5 00101100=>3 00101101=>5 00101110=>5 00101111=>7 00110000=>4 00110001=>4 00110010=>3 00110011=>5 00110100=>3 00110101=>4 00110110=>4 00110111=>6 00111000=>3 00111001=>4 00111010=>4 00111011=>6 00111100=>4 00111101=>6 00111110=>6 00111111=>8 01000000=>6 01000001=>6 01000010=>5 01000011=>6 01000100=>4 01000101=>5 01000110=>4 01000111=>6 01001000=>4 01001001=>4 01001010=>3 01001011=>5 01001100=>3 01001101=>4 01001110=>4 01001111=>6 01010000=>4 01010001=>4 01010010=>3 01010011=>4 01010100=>2 01010101=>3 01010110=>3 01010111=>5 01011000=>2 01011001=>3 01011010=>3 01011011=>5 01011100=>3 01011101=>5 01011110=>5 01011111=>7 01100000=>4 01100001=>4 01100010=>3 01100011=>4 01100100=>2 01100101=>3 01100110=>3 01100111=>5 01101000=>2 01101001=>3 01101010=>2 01101011=>4 01101100=>2 01101101=>4 01101110=>4 01101111=>6 01110000=>2 01110001=>3 01110010=>2 01110011=>4 01110100=>2 01110101=>4 01110110=>4 01110111=>6 01111000=>2 01111001=>4 01111010=>4 01111011=>6 01111100=>4 01111101=>6 01111110=>6 01111111=>8 10000000=>6 10000001=>6 10000010=>5 10000011=>6 10000100=>4 10000101=>5 10000110=>4 10000111=>6 10001000=>4 10001001=>4 10001010=>3 10001011=>5 10001100=>3 10001101=>4 10001110=>4 10001111=>6 10010000=>4 10010001=>4 10010010=>3 10010011=>4 10010100=>2 10010101=>3 10010110=>3 10010111=>5 10011000=>2 10011001=>3 10011010=>2 10011011=>4 10011100=>2 10011101=>4 10011110=>4 10011111=>6 10100000=>4 10100001=>4 10100010=>3 10100011=>4 10100100=>2 10100101=>3 10100110=>2 10100111=>4 10101000=>2 10101001=>2 10101010=>1 10101011=>3 10101100=>1 10101101=>3 10101110=>3 10101111=>5 10110000=>2 10110001=>2 10110010=>1 10110011=>3 10110100=>1 10110101=>3 10110110=>3 10110111=>5 10111000=>1 10111001=>3 10111010=>3 10111011=>5 10111100=>3 10111101=>5 10111110=>5 10111111=>7 11000000=>4 11000001=>4 11000010=>3 11000011=>4 11000100=>2 11000101=>3 11000110=>2 11000111=>4 11001000=>2 11001001=>2 11001010=>1 11001011=>3 11001100=>1 11001101=>3 11001110=>3 11001111=>5 11010000=>2 11010001=>2 11010010=>1 11010011=>3 11010100=>1 11010101=>2 11010110=>2 11010111=>4 11011000=>1 11011001=>2 11011010=>2 11011011=>4 11011100=>2 11011101=>4 11011110=>4 11011111=>6 11100000=>2 11100001=>2 11100010=>1 11100011=>3 11100100=>1 11100101=>2 11100110=>2 11100111=>4 11101000=>1 11101001=>2 11101010=>2 11101011=>4 11101100=>2 11101101=>4 11101110=>4 11101111=>6 11110000=>1 11110001=>2 11110010=>2 11110011=>4 11110100=>2 11110101=>4 11110110=>4 11110111=>6 11111000=>2 11111001=>4 11111010=>4 11111011=>6 11111100=>4 11111101=>6 11111110=>6 11111111=>8 000000000=>9 000000001=>9 000000010=>8 000000011=>9 000000100=>7 000000101=>8 000000110=>7 000000111=>9 000001000=>7 000001001=>7 000001010=>6 000001011=>8 000001100=>6 000001101=>7 000001110=>7 000001111=>9 000010000=>7 000010001=>7 000010010=>6 000010011=>7 000010100=>5 000010101=>6 000010110=>6 000010111=>8 000011000=>5 000011001=>6 000011010=>5 000011011=>7 000011100=>5 000011101=>7 000011110=>7 000011111=>9 000100000=>7 000100001=>7 000100010=>6 000100011=>7 000100100=>5 000100101=>6 000100110=>5 000100111=>7 000101000=>5 000101001=>5 000101010=>4 000101011=>6 000101100=>4 000101101=>6 000101110=>6 000101111=>8 000110000=>5 000110001=>5 000110010=>4 000110011=>6 000110100=>4 000110101=>5 000110110=>5 000110111=>7 000111000=>4 000111001=>5 000111010=>5 000111011=>7 000111100=>5 000111101=>7 000111110=>7 000111111=>9 001000000=>7 001000001=>7 001000010=>6 001000011=>7 001000100=>5 001000101=>6 001000110=>5 001000111=>7 001001000=>5 001001001=>5 001001010=>4 001001011=>6 001001100=>4 001001101=>5 001001110=>5 001001111=>7 001010000=>5 001010001=>5 001010010=>4 001010011=>5 001010100=>3 001010101=>4 001010110=>4 001010111=>6 001011000=>3 001011001=>4 001011010=>4 001011011=>6 001011100=>4 001011101=>6 001011110=>6 001011111=>8 001100000=>5 001100001=>5 001100010=>4 001100011=>5 001100100=>3 001100101=>4 001100110=>4 001100111=>6 001101000=>3 001101001=>4 001101010=>3 001101011=>5 001101100=>3 001101101=>5 001101110=>5 001101111=>7 001110000=>3 001110001=>4 001110010=>3 001110011=>5 001110100=>3 001110101=>5 001110110=>5 001110111=>7 001111000=>3 001111001=>5 001111010=>5 001111011=>7 001111100=>5 001111101=>7 001111110=>7 001111111=>9 010000000=>7 010000001=>7 010000010=>6 010000011=>7 010000100=>5 010000101=>6 010000110=>5 010000111=>7 010001000=>5 010001001=>5 010001010=>4 010001011=>6 010001100=>4 010001101=>5 010001110=>5 010001111=>7 010010000=>5 010010001=>5 010010010=>4 010010011=>5 010010100=>3 010010101=>4 010010110=>4 010010111=>6 010011000=>3 010011001=>4 010011010=>3 010011011=>5 010011100=>3 010011101=>5 010011110=>5 010011111=>7 010100000=>5 010100001=>5 010100010=>4 010100011=>5 010100100=>3 010100101=>4 010100110=>3 010100111=>5 010101000=>3 010101001=>3 010101010=>2 010101011=>4 010101100=>2 010101101=>4 010101110=>4 010101111=>6 010110000=>3 010110001=>3 010110010=>2 010110011=>4 010110100=>2 010110101=>4 010110110=>4 010110111=>6 010111000=>2 010111001=>4 010111010=>4 010111011=>6 010111100=>4 010111101=>6 010111110=>6 010111111=>8 011000000=>5 011000001=>5 011000010=>4 011000011=>5 011000100=>3 011000101=>4 011000110=>3 011000111=>5 011001000=>3 011001001=>3 011001010=>2 011001011=>4 011001100=>2 011001101=>4 011001110=>4 011001111=>6 011010000=>3 011010001=>3 011010010=>2 011010011=>4 011010100=>2 011010101=>3 011010110=>3 011010111=>5 011011000=>2 011011001=>3 011011010=>3 011011011=>5 011011100=>3 011011101=>5 011011110=>5 011011111=>7 011100000=>3 011100001=>3 011100010=>2 011100011=>4 011100100=>2 011100101=>3 011100110=>3 011100111=>5 011101000=>2 011101001=>3 011101010=>3 011101011=>5 011101100=>3 011101101=>5 011101110=>5 011101111=>7 011110000=>2 011110001=>3 011110010=>3 011110011=>5 011110100=>3 011110101=>5 011110110=>5 011110111=>7 011111000=>3 011111001=>5 011111010=>5 011111011=>7 011111100=>5 011111101=>7 011111110=>7 011111111=>9 100000000=>7 100000001=>7 100000010=>6 100000011=>7 100000100=>5 100000101=>6 100000110=>5 100000111=>7 100001000=>5 100001001=>5 100001010=>4 100001011=>6 100001100=>4 100001101=>5 100001110=>5 100001111=>7 100010000=>5 100010001=>5 100010010=>4 100010011=>5 100010100=>3 100010101=>4 100010110=>4 100010111=>6 100011000=>3 100011001=>4 100011010=>3 100011011=>5 100011100=>3 100011101=>5 100011110=>5 100011111=>7 100100000=>5 100100001=>5 100100010=>4 100100011=>5 100100100=>3 100100101=>4 100100110=>3 100100111=>5 100101000=>3 100101001=>3 100101010=>2 100101011=>4 100101100=>2 100101101=>4 100101110=>4 100101111=>6 100110000=>3 100110001=>3 100110010=>2 100110011=>4 100110100=>2 100110101=>3 100110110=>3 100110111=>5 100111000=>2 100111001=>3 100111010=>3 100111011=>5 100111100=>3 100111101=>5 100111110=>5 100111111=>7 101000000=>5 101000001=>5 101000010=>4 101000011=>5 101000100=>3 101000101=>4 101000110=>3 101000111=>5 101001000=>3 101001001=>3 101001010=>2 101001011=>4 101001100=>2 101001101=>3 101001110=>3 101001111=>5 101010000=>3 101010001=>3 101010010=>2 101010011=>3 101010100=>1 101010101=>2 101010110=>2 101010111=>4 101011000=>1 101011001=>2 101011010=>2 101011011=>4 101011100=>2 101011101=>4 101011110=>4 101011111=>6 101100000=>3 101100001=>3 101100010=>2 101100011=>3 101100100=>1 101100101=>2 101100110=>2 101100111=>4 101101000=>1 101101001=>2 101101010=>2 101101011=>4 101101100=>2 101101101=>4 101101110=>4 101101111=>6 101110000=>1 101110001=>2 101110010=>2 101110011=>4 101110100=>2 101110101=>4 101110110=>4 101110111=>6 101111000=>2 101111001=>4 101111010=>4 101111011=>6 101111100=>4 101111101=>6 101111110=>6 101111111=>8 110000000=>5 110000001=>5 110000010=>4 110000011=>5 110000100=>3 110000101=>4 110000110=>3 110000111=>5 110001000=>3 110001001=>3 110001010=>2 110001011=>4 110001100=>2 110001101=>3 110001110=>3 110001111=>5 110010000=>3 110010001=>3 110010010=>2 110010011=>3 110010100=>1 110010101=>2 110010110=>2 110010111=>4 110011000=>1 110011001=>2 110011010=>2 110011011=>4 110011100=>2 110011101=>4 110011110=>4 110011111=>6 110100000=>3 110100001=>3 110100010=>2 110100011=>3 110100100=>1 110100101=>2 110100110=>2 110100111=>4 110101000=>1 110101001=>2 110101010=>1 110101011=>3 110101100=>1 110101101=>3 110101110=>3 110101111=>5 110110000=>1 110110001=>2 110110010=>1 110110011=>3 110110100=>1 110110101=>3 110110110=>3 110110111=>5 110111000=>1 110111001=>3 110111010=>3 110111011=>5 110111100=>3 110111101=>5 110111110=>5 110111111=>7 111000000=>3 111000001=>3 111000010=>2 111000011=>3 111000100=>1 111000101=>2 111000110=>2 111000111=>4 111001000=>1 111001001=>2 111001010=>1 111001011=>3 111001100=>1 111001101=>3 111001110=>3 111001111=>5 111010000=>1 111010001=>2 111010010=>1 111010011=>3 111010100=>1 111010101=>3 111010110=>3 111010111=>5 111011000=>1 111011001=>3 111011010=>3 111011011=>5 111011100=>3 111011101=>5 111011110=>5 111011111=>7 111100000=>1 111100001=>2 111100010=>1 111100011=>3 111100100=>1 111100101=>3 111100110=>3 111100111=>5 111101000=>1 111101001=>3 111101010=>3 111101011=>5 111101100=>3 111101101=>5 111101110=>5 111101111=>7 111110000=>1 111110001=>3 111110010=>3 111110011=>5 111110100=>3 111110101=>5 111110110=>5 111110111=>7 111111000=>3 111111001=>5 111111010=>5 111111011=>7 111111100=>5 111111101=>7 111111110=>7 111111111=>9 1010101010=>1 1010101100=>1 1010110010=>1 1010110100=>1 1010111000=>1 1011001010=>1 1011001100=>1 1011010010=>1 1011010100=>1 1011011000=>1 1011100010=>1 1011100100=>1 1011101000=>1 1011110000=>1 1100101010=>1 1100101100=>1 1100110010=>1 1100110100=>1 1100111000=>1 1101001010=>1 1101001100=>1 1101010010=>1 1101010100=>1 1101011000=>1 1101100010=>1 1101100100=>1 1101101000=>1 1101110000=>1 1110001010=>1 1110001100=>1 1110010010=>1 1110010100=>1 1110011000=>1 1110100010=>1 1110100100=>1 1110101000=>1 1110110000=>1 1111000010=>1 1111000100=>1 1111001000=>1 1111010000=>1 1111100000=>1 1111111110=>8 1111111101=>8 1111000001=>2 1110111110=>6 1110100001=>2 1110011110=>4 1110010001=>2 1110001001=>2 1110000101=>3 1101111110=>6 1101100001=>2 1101010001=>2 1101001110=>3 1101001001=>2 1101000101=>3 1100110001=>2 1100101001=>2 1100100110=>2 1100100101=>3 1100011001=>3 1100010101=>3 1100010010=>3 1100001000=>4 1011111110=>7 1011111101=>7 1011100001=>2 1011010001=>2 1011001001=>2 1011000101=>3 1010111110=>5 1010110001=>2 1010101001=>2 1010100110=>2 1010100101=>3 1010011010=>2 1010011001=>3 1010010110=>3 1010010101=>3 1010001110=>4 1001111110=>6 1001111000=>2 1001110100=>2 1001110001=>3 1001101100=>2 1001101010=>2 1001101001=>3 1001100110=>3 1001100101=>3 1001011110=>5 1001011010=>3 1001011001=>3 1001010110=>3 1001010101=>3 1001001110=>4 1000111110=>6 1000111100=>4 1000111010=>4 1000110110=>4 1000101110=>5 1000011110=>6 1000010110=>5 1000001110=>6 1000001010=>5 1000000110=>6 1000000100=>6 1000000010=>7 1000000000=>8 0111111111=>10 0111111101=>8 0111111001=>6 0111110001=>4 0111100001=>3 0111000001=>4 0110000001=>6 0100000001=>8 0000000001=>10 0111111011=>8 0111110101=>6 0111101001=>4 0111010001=>3 0110100001=>4 0101000001=>6 0010000001=>8 0101010101=>3 0101010011=>4 0101001101=>4 0101001011=>5 0101000111=>6 0100110101=>4 0100110011=>5 0100101101=>5 0100101011=>5 0100100111=>6 0100011101=>6 0100011011=>6 0100010111=>7 0100001111=>8 0011010101=>4 0011010011=>5 0011001101=>5 0011001011=>5 0011000111=>6 0010110101=>5 0010110011=>5 0010101101=>5 0010101011=>5 0010100111=>6 0010011101=>6 0010011011=>6 0010010111=>7 0010001111=>8 0001110101=>6 0001110011=>6 0001101101=>6 0001101011=>6 0001100111=>7 0001011101=>7 0001011011=>7 0001010111=>7 0001001111=>8 0000111101=>8 0000111011=>8 0000110111=>8 0000101111=>9 0000011111=>10 0000000000=>10 0000000110=>8 0000011000=>6 0000011110=>8 0000010010=>7 0001100000=>6 0001100110=>5 0001111000=>4 0001111110=>8 0001110010=>4 0001001000=>6 0001001110=>6 0001000010=>7 0001011010=>5 0100000010=>7 0010101010=>3 0010101000=>4 0010101100=>3 0010100100=>4 0010010100=>4 0001010100=>4 0010101110=>5 0010100000=>6 0010111100=>5 0010000100=>6 0011110100=>4 0000010100=>6 0111010100=>2 0000000010=>9 0000001110=>8 0000111110=>8 0000000100=>8 0000001010=>7 0000010110=>7 0000101110=>7 0001011110=>7 0100000000=>8 0100000011=>8 0100000111=>8 0100011111=>8 0100111111=>8 0101111111=>9 0100000101=>7 0100001011=>7 0100101111=>7 0101011111=>7 0110111111=>8 0101010110=>3 0101011001=>3 0101011010=>3 0101011100=>3 0101100101=>3 0101100110=>3 0101101001=>3 0101101010=>3 0101101100=>3 0101110001=>3 0101110010=>3 0101110100=>3 0101111000=>3 0110010101=>3 0110010110=>3 0110011001=>3 0110011010=>3 0110011100=>3 0110100101=>3 0110100110=>3 0110101001=>3 0110101010=>2 0110101100=>2 0110110001=>3 0110110010=>2 0110110100=>2 0110111000=>2 0111000101=>3 0111000110=>3 0111001001=>3 0111001010=>2 0111001100=>2 0111010010=>2 0111011000=>2 0111100010=>2 0111100100=>2 0111101000=>2 0111110000=>2 0100000100=>6 0011100110=>4 0011101010=>4 0011101100=>4 0001101110=>6 0010110110=>5 0011110010=>4 0001110110=>6 0100111010=>4 0010111010=>5 0011111000=>4 0001111010=>6 0100111100=>4 0001111100=>6 0000000101=>9 0000001101=>8 0000011101=>8 0001111101=>8 0011111101=>8 0000001001=>8 0000010101=>7 0000101101=>7 0010111101=>7 0101111101=>7 0010110001=>4 0001100001=>6 0001000001=>8 0000111111=>10 0001110100=>4 0001101100=>4 0010011100=>4 0001011100=>5 0000111100=>6 0001101010=>4 0010011010=>4 0000111010=>6 0010100110=>4 0010010110=>5 0001010110=>5 0000110110=>6 0010001110=>6 0001101001=>5 0000110001=>6 0000010001=>8 0000001100=>7 0000011001=>7 0000110011=>7 0000110100=>5 0000000011=>10 0000011011=>8 0001101111=>8 0000110101=>6 0010000010=>7 0001000011=>8 0010011110=>6 0001010000=>6 0001000100=>6 0001010101=>5 0001001011=>7 0001100011=>6 0001110111=>8 0010000000=>8 0000000111=>10 0000001111=>10 0001111111=>10 0000001011=>9 0000010111=>9 0001011111=>9 0001111001=>6 0000011010=>6 0000101010=>5 0000100101=>7 0010010001=>6 0010000101=>7 0010010101=>5 0010010011=>6 0000101011=>7 0001000111=>8 0000100111=>8 0010010010=>5 0001001010=>5 0001001001=>6 0001000101=>7 0000100011=>8 0000100100=>6 0000100010=>7 0010110000=>4 0000011100=>6 0000001000=>8 0000101100=>5 0010101111=>7 0000101000=>6 0010001000=>6 0001001100=>5 0001000110=>6 0001100100=>4 0001100010=>5 0000100110=>6 0000110010=>5 0010010000=>6 0001010010=>5 0001000000=>8 0001011001=>5 0001100101=>5 0001110001=>5 0010011001=>5 0010100101=>5 0010101001=>4 0010011111=>8 0000111001=>6 0010001101=>6 0001001101=>6 0001010001=>6 0001010011=>6 0001111011=>8 0010011000=>4 0001011000=>4 0000100000=>8 0010001011=>7 0010000111=>8 0000010011=>8 0010001100=>5 0010000110=>6 0000110000=>6 0000100001=>8 0010100011=>6 0010100001=>6 0000111000=>5 0001101000=>4 0000010000=>8 0000101001=>6 0010100010=>5 0010001001=>6 0010000011=>8 0010001010=>5 0001110000=>4 10000000000=>9 10000000010=>8 10000000110=>7 10000001110=>7 10000011110=>7 10000111110=>7 10001111110=>7 10011111110=>7 10111111110=>8 11111111110=>9 10000000100=>7 10000001010=>6 10000010110=>6 10000101110=>6 10001011110=>6 10010111110=>6 10101111110=>6 11011111110=>7 10000010000=>7 10000100010=>6 10001000010=>6 10000100100=>5 10001001010=>4 10001000100=>5 10000101000=>5 10010001010=>4 10001010010=>4 10010000100=>5 10001001000=>5 10010010010=>4 10001010100=>3 10010101010=>2 10010001000=>5 10100010010=>4 10010100010=>4 10010010100=>3 10100101010=>2 01111111111=>11 01111111101=>9 01111111001=>7 01111110001=>5 01111100001=>3 01111000001=>3 01110000001=>5 01100000001=>7 01000000001=>9 00000000001=>11 00000000010=>10 10101010101=>2 10101010010=>2 10101001001=>3 10101001010=>2 10101000100=>3 10100100101=>4 10100100010=>4 10100101001=>3 10100100100=>3 10100010001=>5 10100010100=>3 10100001000=>5 10010010101=>4 10010001001=>5 10010100101=>4 10010101001=>3 10010100100=>3 10010010001=>5 10001000101=>6 10001001001=>5 10001010001=>5 10000100001=>7 01001010101=>4 01001010010=>4 01001001001=>5 01001001010=>4 01001000100=>5 01000100101=>6 01000100010=>6 01000101001=>5 01000101010=>4 01000100100=>5 01000010001=>7 01000010010=>6 01000010100=>5 01000001000=>7 01010010101=>4 01010010010=>4 01010001001=>5 01010001010=>4 01010000100=>5 01010100101=>4 01010100010=>4 01010101001=>3 01010101010=>2 01010100100=>3 01010010001=>5 01010010100=>3 01010001000=>5 01001000101=>6 01001000010=>6 01001010001=>5 01001010100=>3 01001001000=>5 01000100001=>7 01000101000=>5 01000010000=>7 00100010101=>6 00100010010=>6 00100001001=>7 00100001010=>6 00100000100=>7 00100100101=>6 00100100010=>6 00100101001=>5 00100101010=>4 00100100100=>5 00100010001=>7 00100010100=>5 00100001000=>7 00101000101=>6 00101000010=>6 00101001001=>5 00101001010=>4 00101000100=>5 00101010001=>5 00101010010=>4 00101010100=>3 00101001000=>5 00100100001=>7 00100101000=>5 00100010000=>7 00010000101=>8 00010000010=>8 00010001001=>7 00010001010=>6 00010000100=>7 00010010001=>7 00010010010=>6 00010010100=>5 00010001000=>7 00010100001=>7 00010100010=>6 00010100100=>5 00010101000=>5 00010010000=>7 00001000001=>9 00001000010=>8 00001000100=>7 00001001000=>7 00001010000=>7 00000100000=>9 01000000000=>9 01000000011=>9 01000000111=>9 01000001111=>9 01000011111=>9 01000111111=>9 01001111111=>9 01011111111=>10 00000000101=>10 00000001101=>9 00000011101=>9 00000111101=>9 00001111101=>9 00011111101=>9 00111111101=>9 11111111101=>9 101010101010=>1 101010101100=>1 101010110010=>1 101010110100=>1 101010111000=>1 101011001010=>1 101011001100=>1 101011010010=>1 101011010100=>1 101011011000=>1 101011100010=>1 101011100100=>1 101011101000=>1 101011110000=>1 101100101010=>1 101100101100=>1 101100110010=>1 101100110100=>1 101100111000=>1 101101001010=>1 101101001100=>1 101101010010=>1 101101010100=>1 101101011000=>1 101101100010=>1 101101100100=>1 101101101000=>1 101101110000=>1 101110001010=>1 101110001100=>1 101110010010=>1 101110010100=>1 101110011000=>1 101110100010=>1 101110100100=>1 101110101000=>1 101110110000=>1 101111000010=>1 101111000100=>1 101111001000=>1 101111010000=>1 101111100000=>1 110010101010=>1 110010101100=>1 110010110010=>1 110010110100=>1 110010111000=>1 110011001010=>1 110011001100=>1 110011010010=>1 110011010100=>1 110011011000=>1 110011100010=>1 110011100100=>1 110011101000=>1 110011110000=>1 110100101010=>1 110100101100=>1 110100110010=>1 110100110100=>1 110100111000=>1 110101001010=>1 110101001100=>1 110101010010=>1 110101010100=>1 110101011000=>1 110101100010=>1 110101100100=>1 110101101000=>1 110101110000=>1 110110001010=>1 110110001100=>1 110110010010=>1 110110010100=>1 110110011000=>1 110110100010=>1 110110100100=>1 110110101000=>1 110110110000=>1 110111000010=>1 110111000100=>1 110111001000=>1 110111010000=>1 110111100000=>1 111000101010=>1 111000101100=>1 111000110010=>1 111000110100=>1 111000111000=>1 111001001010=>1 111001001100=>1 111001010010=>1 111001010100=>1 111001011000=>1 111001100010=>1 111001100100=>1 111001101000=>1 111001110000=>1 111010001010=>1 111010001100=>1 111010010010=>1 111010010100=>1 111010011000=>1 111010100010=>1 111010100100=>1 111010101000=>1 111010110000=>1 111011000010=>1 111011000100=>1 111011001000=>1 111011010000=>1 111011100000=>1 111100001010=>1 111100001100=>1 111100010010=>1 111100010100=>1 111100011000=>1 111100100010=>1 111100100100=>1 111100101000=>1 111100110000=>1 111101000010=>1 111101000100=>1 111101001000=>1 111101010000=>1 111101100000=>1 111110000010=>1 111110000100=>1 111110001000=>1 111110010000=>1 111110100000=>1 111111000000=>1 100001111100=>6 010101010101=>3 010101010011=>4 010101001101=>4 010101001011=>5 010101000111=>6 010100110101=>4 010100110011=>5 010100101101=>5 010100101011=>5 010100100111=>6 010100011101=>6 010100011011=>6 010100010111=>7 010100001111=>8 010011010101=>4 010011010011=>5 010011001101=>5 010011001011=>5 010011000111=>6 010010110101=>5 010010110011=>5 010010101101=>5 010010101011=>5 010010100111=>6 010010011101=>6 010010011011=>6 010010010111=>7 010010001111=>8 010001110101=>6 010001110011=>6 010001101101=>6 010001101011=>6 010001100111=>7 010001011101=>7 010001011011=>7 010001010111=>7 010001001111=>8 010000111101=>8 010000111011=>8 010000110111=>8 010000101111=>9 010000011111=>10 001101010101=>4 001101010011=>5 001101001101=>5 001101001011=>5 001101000111=>6 001100110101=>5 001100110011=>5 001100101101=>5 001100101011=>5 001100100111=>6 001100011101=>6 001100011011=>6 001100010111=>7 001100001111=>8 001011010101=>5 001011010011=>5 001011001101=>5 001011001011=>5 001011000111=>6 001010110101=>5 001010110011=>5 001010101101=>5 001010101011=>5 001010100111=>6 001010011101=>6 001010011011=>6 001010010111=>7 001010001111=>8 001001110101=>6 001001110011=>6 001001101101=>6 001001101011=>6 001001100111=>7 001001011101=>7 001001011011=>7 001001010111=>7 001001001111=>8 001000111101=>8 001000111011=>8 001000110111=>8 001000101111=>9 001000011111=>10 000111010101=>6 000111010011=>6 000111001101=>6 000111001011=>6 000111000111=>7 000110110101=>6 000110110011=>6 000110101101=>6 000110101011=>6 000110100111=>7 000110011101=>7 000110011011=>7 000110010111=>7 000110001111=>8 000101110101=>7 000101110011=>7 000101101101=>7 000101101011=>7 000101100111=>7 000101011101=>7 000101011011=>7 000101010111=>7 000101001111=>8 000100111101=>8 000100111011=>8 000100110111=>8 000100101111=>9 000100011111=>10 000011110101=>8 000011110011=>8 000011101101=>8 000011101011=>8 000011100111=>8 000011011101=>8 000011011011=>8 000011010111=>8 000011001111=>9 000010111101=>9 000010111011=>9 000010110111=>9 000010101111=>9 000010011111=>10 000001111101=>10 000001111011=>10 000001110111=>10 000001101111=>10 000001011111=>11 000000111111=>12 111110000001=>2 111001100001=>2 111000011001=>3 111000110001=>2 111000100101=>3 111011000001=>2 111010010001=>2 111010000101=>3 100111100001=>3 100110011001=>3 100110110001=>3 100110100101=>3 101101100001=>2 101100011001=>3 101100110001=>2 101100100101=>3 101111000001=>2 101110010001=>2 101110000101=>3 101001011001=>3 101001110001=>3 101001100101=>3 101011010001=>2 101011000101=>3 101010010101=>3 000001111110=>10 000111100110=>6 000111111000=>6 000111110010=>6 000111001110=>6 000111011010=>6 000101101110=>7 000101111010=>7 011001100110=>3 011001111000=>3 011001110010=>3 011001011010=>3 011110000110=>3 011110011000=>2 011110010010=>2 011111100000=>2 011111001000=>2 011111000010=>2 011100100110=>3 011100111000=>2 011100110010=>2 011100011010=>3 011101101000=>2 011101100010=>2 011101001010=>2 010011111000=>4 010011110010=>4 010011011010=>4 010001111010=>6 010110100110=>3 010110111000=>3 010110110010=>3 010110011010=>3 010111101000=>3 010111100010=>3 010111001010=>3 010101101010=>3 010101010110=>3 010101011001=>3 010101011010=>3 010101011100=>3 010101100101=>3 010101100110=>3 010101101001=>3 010101101100=>3 010101110001=>3 010101110010=>3 010101110100=>3 010101111000=>3 010110010101=>3 010110010110=>3 010110011001=>3 010110011100=>3 010110100101=>3 010110101001=>3 010110101010=>3 010110101100=>3 010110110001=>3 010110110100=>3 010111000101=>3 010111000110=>3 010111001001=>3 010111001100=>3 010111010001=>3 010111010010=>3 010111010100=>3 010111011000=>3 010111100001=>3 010111100100=>3 010111110000=>3 011001010101=>3 011001010110=>3 011001011001=>3 011001011100=>3 011001100101=>3 011001101001=>3 011001101010=>3 011001101100=>3 011001110001=>3 011001110100=>3 011010010101=>3 011010010110=>3 011010011001=>3 011010011010=>3 011010011100=>3 011010100101=>3 011010100110=>3 011010101001=>3 011010101010=>2 011010101100=>2 011010110001=>3 011010110010=>2 011010110100=>2 011010111000=>2 011011000101=>3 011011000110=>3 011011001001=>3 011011001010=>2 011011001100=>2 011011010001=>3 011011010010=>2 011011010100=>2 011011011000=>2 011011100001=>3 011011100010=>2 011011100100=>2 011011101000=>2 011011110000=>2 011100010101=>3 011100010110=>3 011100011001=>3 011100011100=>3 011100100101=>3 011100101001=>3 011100101010=>2 011100101100=>2 011100110001=>3 011100110100=>2 011101000101=>3 011101000110=>3 011101001001=>3 011101001100=>2 011101010001=>3 011101010010=>2 011101010100=>2 011101011000=>2 011101100001=>3 011101100100=>2 011101110000=>2 011110000101=>3 011110001001=>3 011110001010=>2 011110001100=>2 011110010001=>3 011110010100=>2 011110100001=>3 011110100010=>2 011110100100=>2 011110101000=>2 011110110000=>2 011111000001=>3 011111000100=>2 011111010000=>2 001111000110=>4 001111001010=>4 100111001100=>2 001111001100=>4 001011010110=>5 001111010010=>4 100111010100=>2 001111010100=>4 000111010110=>6 001011011010=>5 100111011000=>2 001111011000=>4 100011011100=>4 010011011100=>4 001011011100=>5 000111011100=>6 000011011110=>8 001101100110=>4 001101101010=>4 100101101100=>3 001101101100=>4 001011100110=>5 001111100010=>4 100111100100=>2 001111100100=>4 010011101010=>4 001011101010=>5 100111101000=>2 001111101000=>4 000111101010=>6 100011101100=>4 010011101100=>4 001011101100=>5 000111101100=>6 000011101110=>8 101001110100=>2 001001110110=>6 001101110010=>4 100101110100=>3 001101110100=>4 000101110110=>7 001011110010=>5 100111110000=>2 001111110000=>4 100011110100=>4 010011110100=>4 001011110100=>5 000111110100=>6 000011110110=>8 110001111000=>2 101001111000=>2 001001111010=>6 100101111000=>3 001101111000=>4 100011111000=>4 001011111000=>5 000011111010=>8 010001111100=>6 001001111100=>6 000101111100=>7 000011111100=>8 100101010101=>3 100101011001=>3 100101100101=>3 100101101001=>3 100101110001=>3 100110010101=>3 100110101001=>3 100111000101=>3 100111001001=>3 100111010001=>3 101001010101=>3 101001101001=>3 101010011001=>3 101010100101=>3 101010101001=>2 101010110001=>2 101011001001=>2 101011100001=>2 101100010101=>3 101100101001=>2 101101000101=>3 101101001001=>2 101101010001=>2 101110001001=>2 101110100001=>2 110001010101=>3 110001011001=>3 110001100101=>3 110001101001=>3 110001110001=>3 110010010101=>3 110010011001=>3 110010100101=>3 110010101001=>2 110010110001=>2 110011000101=>3 110011001001=>2 110011010001=>2 110011100001=>2 110100010101=>3 110100011001=>3 110100100101=>3 110100101001=>2 110100110001=>2 110101000101=>3 110101001001=>2 110101010001=>2 110101100001=>2 110110000101=>3 110110001001=>2 110110010001=>2 110110100001=>2 110111000001=>2 111000010101=>3 111000101001=>2 111001000101=>3 111001001001=>2 111001010001=>2 111010001001=>2 111010100001=>2 111100000101=>3 111100001001=>2 111100010001=>2 111100100001=>2 111101000001=>2 10110111100000=>1 10111011010000=>1 10111011100000=>1 10111100110000=>1 10111101001000=>1 10111101010000=>1 10111101100000=>1 10111110000100=>1 10111110001000=>1 10111110010000=>1 10111110100000=>1 10111111000000=>1 11001111100000=>1 11011110000010=>1 11101101000010=>1 11101110000010=>1 11110011000010=>1 11110100100010=>1 11110101000010=>1 11110110000010=>1 11111000001100=>1 11111000010010=>1 11111000100010=>1 11111001000010=>1 11111010000010=>1 11111100000010=>1 1011110111000000=>1 1011111010100000=>1 1011111011000000=>1 1011111100010000=>1 1011111100100000=>1 1011111101000000=>1 1011111110000000=>1 1111011100000010=>1 1111101010000010=>1 1111101100000010=>1 1111110001000010=>1 1111110010000010=>1 1111110100000010=>1 1111111000000010=>1 111111110000000010=>1 101111111100000000=>1 111111101000000010=>1 101111111010000000=>1 111111100100000010=>1 111111011000000010=>1 101111111001000000=>1 101111110110000000=>1 11111111100000000010=>1 10111111111000000000=>1 11111111010000000010=>1 10111111110100000000=>1 1111111111000000000010=>1 1011111111110000000000=>1
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Description
The number of factors in the Catalan decomposition 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 number of factors in the Catalan factorisation, that is, $\ell + m$ if the middle Dyck word is empty and $\ell + 1 + m$ otherwise.
References
[1] Lothaire, M. Applied combinatorics on words MathSciNet:2165687
Code
def statistic(w):
    c = catalan_factorization(w)
    if len(c[1]) == 0:
        return len(c[0]) + len(c[2])
    else:
        return len(c[0]) + 1 + len(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 08:18 by Martin Rubey
Updated
Dec 28, 2017 at 15:24 by Martin Rubey