edit this statistic or download as text // json
Identifier
Values
=>
0=>1 1=>1 00=>1 01=>2 10=>2 11=>1 000=>1 001=>2 010=>1 011=>2 100=>2 101=>1 110=>2 111=>1 0000=>1 0001=>2 0010=>2 0011=>2 0100=>2 0101=>2 0110=>1 0111=>2 1000=>2 1001=>1 1010=>2 1011=>2 1100=>2 1101=>2 1110=>2 1111=>1 00000=>1 00001=>2 00010=>2 00011=>2 00100=>1 00101=>2 00110=>2 00111=>2 01000=>2 01001=>2 01010=>1 01011=>2 01100=>2 01101=>2 01110=>1 01111=>2 10000=>2 10001=>1 10010=>2 10011=>2 10100=>2 10101=>1 10110=>2 10111=>2 11000=>2 11001=>2 11010=>2 11011=>1 11100=>2 11101=>2 11110=>2 11111=>1 000000=>1 000001=>2 000010=>2 000011=>2 000100=>2 000101=>2 000110=>2 000111=>2 001000=>2 001001=>2 001010=>2 001011=>3 001100=>1 001101=>3 001110=>2 001111=>2 010000=>2 010001=>2 010010=>1 010011=>3 010100=>2 010101=>2 010110=>3 010111=>2 011000=>2 011001=>3 011010=>3 011011=>2 011100=>2 011101=>2 011110=>1 011111=>2 100000=>2 100001=>1 100010=>2 100011=>2 100100=>2 100101=>3 100110=>3 100111=>2 101000=>2 101001=>3 101010=>2 101011=>2 101100=>3 101101=>1 101110=>2 101111=>2 110000=>2 110001=>2 110010=>3 110011=>1 110100=>3 110101=>2 110110=>2 110111=>2 111000=>2 111001=>2 111010=>2 111011=>2 111100=>2 111101=>2 111110=>2 111111=>1 0000000=>1 0000001=>2 0000010=>2 0000011=>2 0000100=>2 0000101=>2 0000110=>2 0000111=>2 0001000=>1 0001001=>2 0001010=>2 0001011=>3 0001100=>2 0001101=>3 0001110=>2 0001111=>2 0010000=>2 0010001=>2 0010010=>2 0010011=>2 0010100=>1 0010101=>2 0010110=>3 0010111=>3 0011000=>2 0011001=>2 0011010=>3 0011011=>2 0011100=>1 0011101=>3 0011110=>2 0011111=>2 0100000=>2 0100001=>2 0100010=>1 0100011=>3 0100100=>2 0100101=>2 0100110=>2 0100111=>3 0101000=>2 0101001=>2 0101010=>1 0101011=>2 0101100=>3 0101101=>2 0101110=>3 0101111=>2 0110000=>2 0110001=>3 0110010=>2 0110011=>2 0110100=>3 0110101=>2 0110110=>1 0110111=>2 0111000=>2 0111001=>3 0111010=>3 0111011=>2 0111100=>2 0111101=>2 0111110=>1 0111111=>2 1000000=>2 1000001=>1 1000010=>2 1000011=>2 1000100=>2 1000101=>3 1000110=>3 1000111=>2 1001000=>2 1001001=>1 1001010=>2 1001011=>3 1001100=>2 1001101=>2 1001110=>3 1001111=>2 1010000=>2 1010001=>3 1010010=>2 1010011=>3 1010100=>2 1010101=>1 1010110=>2 1010111=>2 1011000=>3 1011001=>2 1011010=>2 1011011=>2 1011100=>3 1011101=>1 1011110=>2 1011111=>2 1100000=>2 1100001=>2 1100010=>3 1100011=>1 1100100=>2 1100101=>3 1100110=>2 1100111=>2 1101000=>3 1101001=>3 1101010=>2 1101011=>1 1101100=>2 1101101=>2 1101110=>2 1101111=>2 1110000=>2 1110001=>2 1110010=>3 1110011=>2 1110100=>3 1110101=>2 1110110=>2 1110111=>1 1111000=>2 1111001=>2 1111010=>2 1111011=>2 1111100=>2 1111101=>2 1111110=>2 1111111=>1 00000000=>1 00000001=>2 00000010=>2 00000011=>2 00000100=>2 00000101=>2 00000110=>2 00000111=>2 00001000=>2 00001001=>2 00001010=>2 00001011=>3 00001100=>2 00001101=>3 00001110=>2 00001111=>2 00010000=>2 00010001=>2 00010010=>2 00010011=>3 00010100=>2 00010101=>2 00010110=>3 00010111=>3 00011000=>1 00011001=>3 00011010=>3 00011011=>2 00011100=>2 00011101=>3 00011110=>2 00011111=>2 00100000=>2 00100001=>2 00100010=>2 00100011=>3 00100100=>1 00100101=>2 00100110=>3 00100111=>2 00101000=>2 00101001=>2 00101010=>2 00101011=>3 00101100=>4 00101101=>2 00101110=>3 00101111=>3 00110000=>2 00110001=>3 00110010=>3 00110011=>2 00110100=>4 00110101=>3 00110110=>2 00110111=>3 00111000=>2 00111001=>2 00111010=>3 00111011=>3 00111100=>1 00111101=>3 00111110=>2 00111111=>2 01000000=>2 01000001=>2 01000010=>1 01000011=>3 01000100=>2 01000101=>2 01000110=>3 01000111=>3 01001000=>2 01001001=>2 01001010=>2 01001011=>2 01001100=>3 01001101=>3 01001110=>2 01001111=>3 01010000=>2 01010001=>2 01010010=>2 01010011=>3 01010100=>2 01010101=>2 01010110=>3 01010111=>2 01011000=>3 01011001=>3 01011010=>1 01011011=>2 01011100=>3 01011101=>2 01011110=>3 01011111=>2 01100000=>2 01100001=>3 01100010=>3 01100011=>2 01100100=>3 01100101=>3 01100110=>1 01100111=>3 01101000=>3 01101001=>2 01101010=>3 01101011=>2 01101100=>2 01101101=>2 01101110=>3 01101111=>2 01110000=>2 01110001=>3 01110010=>2 01110011=>3 01110100=>3 01110101=>2 01110110=>3 01110111=>2 01111000=>2 01111001=>3 01111010=>3 01111011=>2 01111100=>2 01111101=>2 01111110=>1 01111111=>2 10000000=>2 10000001=>1 10000010=>2 10000011=>2 10000100=>2 10000101=>3 10000110=>3 10000111=>2 10001000=>2 10001001=>3 10001010=>2 10001011=>3 10001100=>3 10001101=>2 10001110=>3 10001111=>2 10010000=>2 10010001=>3 10010010=>2 10010011=>2 10010100=>2 10010101=>3 10010110=>2 10010111=>3 10011000=>3 10011001=>1 10011010=>3 10011011=>3 10011100=>2 10011101=>3 10011110=>3 10011111=>2 10100000=>2 10100001=>3 10100010=>2 10100011=>3 10100100=>2 10100101=>1 10100110=>3 10100111=>3 10101000=>2 10101001=>3 10101010=>2 10101011=>2 10101100=>3 10101101=>2 10101110=>2 10101111=>2 10110000=>3 10110001=>2 10110010=>3 10110011=>3 10110100=>2 10110101=>2 10110110=>2 10110111=>2 10111000=>3 10111001=>3 10111010=>2 10111011=>2 10111100=>3 10111101=>1 10111110=>2 10111111=>2 11000000=>2 11000001=>2 11000010=>3 11000011=>1 11000100=>3 11000101=>3 11000110=>2 11000111=>2 11001000=>3 11001001=>2 11001010=>3 11001011=>4 11001100=>2 11001101=>3 11001110=>3 11001111=>2 11010000=>3 11010001=>3 11010010=>2 11010011=>4 11010100=>3 11010101=>2 11010110=>2 11010111=>2 11011000=>2 11011001=>3 11011010=>2 11011011=>1 11011100=>3 11011101=>2 11011110=>2 11011111=>2 11100000=>2 11100001=>2 11100010=>3 11100011=>2 11100100=>2 11100101=>3 11100110=>3 11100111=>1 11101000=>3 11101001=>3 11101010=>2 11101011=>2 11101100=>3 11101101=>2 11101110=>2 11101111=>2 11110000=>2 11110001=>2 11110010=>3 11110011=>2 11110100=>3 11110101=>2 11110110=>2 11110111=>2 11111000=>2 11111001=>2 11111010=>2 11111011=>2 11111100=>2 11111101=>2 11111110=>2 11111111=>1 000000000=>1 000000001=>2 000000010=>2 000000011=>2 000000100=>2 000000101=>2 000000110=>2 000000111=>2 000001000=>2 000001001=>2 000001010=>2 000001011=>3 000001100=>2 000001101=>3 000001110=>2 000001111=>2 000010000=>1 000010001=>2 000010010=>2 000010011=>3 000010100=>2 000010101=>2 000010110=>3 000010111=>3 000011000=>2 000011001=>3 000011010=>3 000011011=>2 000011100=>2 000011101=>3 000011110=>2 000011111=>2 000100000=>2 000100001=>2 000100010=>2 000100011=>2 000100100=>2 000100101=>3 000100110=>3 000100111=>3 000101000=>1 000101001=>3 000101010=>2 000101011=>3 000101100=>4 000101101=>2 000101110=>3 000101111=>3 000110000=>2 000110001=>2 000110010=>3 000110011=>2 000110100=>4 000110101=>3 000110110=>2 000110111=>3 000111000=>1 000111001=>3 000111010=>3 000111011=>3 000111100=>2 000111101=>3 000111110=>2 000111111=>2 001000000=>2 001000001=>2 001000010=>2 001000011=>3 001000100=>1 001000101=>3 001000110=>2 001000111=>3 001001000=>2 001001001=>2 001001010=>3 001001011=>3 001001100=>3 001001101=>3 001001110=>3 001001111=>2 001010000=>2 001010001=>3 001010010=>3 001010011=>2 001010100=>1 001010101=>2 001010110=>3 001010111=>3 001011000=>4 001011001=>3 001011010=>2 001011011=>3 001011100=>4 001011101=>2 001011110=>3 001011111=>3 001100000=>2 001100001=>3 001100010=>2 001100011=>2 001100100=>3 001100101=>2 001100110=>2 001100111=>2 001101000=>4 001101001=>3 001101010=>3 001101011=>2 001101100=>1 001101101=>3 001101110=>3 001101111=>3 001110000=>2 001110001=>3 001110010=>3 001110011=>2 001110100=>4 001110101=>3 001110110=>3 001110111=>2 001111000=>2 001111001=>2 001111010=>3 001111011=>3 001111100=>1 001111101=>3 001111110=>2 001111111=>2 010000000=>2 010000001=>2 010000010=>1 010000011=>3 010000100=>2 010000101=>2 010000110=>3 010000111=>3 010001000=>2 010001001=>3 010001010=>3 010001011=>2 010001100=>2 010001101=>3 010001110=>3 010001111=>3 010010000=>2 010010001=>3 010010010=>1 010010011=>3 010010100=>3 010010101=>2 010010110=>3 010010111=>2 010011000=>3 010011001=>2 010011010=>4 010011011=>3 010011100=>3 010011101=>3 010011110=>2 010011111=>3 010100000=>2 010100001=>2 010100010=>3 010100011=>3 010100100=>3 010100101=>2 010100110=>2 010100111=>3 010101000=>2 010101001=>2 010101010=>1 010101011=>2 010101100=>3 010101101=>2 010101110=>3 010101111=>2 010110000=>3 010110001=>3 010110010=>4 010110011=>2 010110100=>2 010110101=>2 010110110=>3 010110111=>3 010111000=>3 010111001=>3 010111010=>1 010111011=>3 010111100=>3 010111101=>2 010111110=>3 010111111=>2 011000000=>2 011000001=>3 011000010=>3 011000011=>2 011000100=>2 011000101=>3 011000110=>1 011000111=>3 011001000=>3 011001001=>3 011001010=>2 011001011=>3 011001100=>2 011001101=>2 011001110=>2 011001111=>3 011010000=>3 011010001=>2 011010010=>3 011010011=>3 011010100=>3 011010101=>2 011010110=>1 011010111=>3 011011000=>2 011011001=>3 011011010=>3 011011011=>2 011011100=>3 011011101=>3 011011110=>3 011011111=>2 011100000=>2 011100001=>3 011100010=>3 011100011=>3 011100100=>3 011100101=>3 011100110=>2 011100111=>2 011101000=>3 011101001=>2 011101010=>3 011101011=>3 011101100=>3 011101101=>3 011101110=>1 011101111=>2 011110000=>2 011110001=>3 011110010=>2 011110011=>3 011110100=>3 011110101=>2 011110110=>3 011110111=>2 011111000=>2 011111001=>3 011111010=>3 011111011=>2 011111100=>2 011111101=>2 011111110=>1 011111111=>2 100000000=>2 100000001=>1 100000010=>2 100000011=>2 100000100=>2 100000101=>3 100000110=>3 100000111=>2 100001000=>2 100001001=>3 100001010=>2 100001011=>3 100001100=>3 100001101=>2 100001110=>3 100001111=>2 100010000=>2 100010001=>1 100010010=>3 100010011=>3 100010100=>3 100010101=>3 100010110=>2 100010111=>3 100011000=>2 100011001=>2 100011010=>3 100011011=>3 100011100=>3 100011101=>3 100011110=>3 100011111=>2 100100000=>2 100100001=>3 100100010=>3 100100011=>3 100100100=>2 100100101=>3 100100110=>3 100100111=>2 100101000=>3 100101001=>1 100101010=>2 100101011=>3 100101100=>3 100101101=>3 100101110=>2 100101111=>3 100110000=>3 100110001=>2 100110010=>2 100110011=>2 100110100=>3 100110101=>2 100110110=>3 100110111=>3 100111000=>3 100111001=>1 100111010=>3 100111011=>2 100111100=>2 100111101=>3 100111110=>3 100111111=>2 101000000=>2 101000001=>3 101000010=>2 101000011=>3 101000100=>3 101000101=>1 101000110=>3 101000111=>3 101001000=>3 101001001=>3 101001010=>2 101001011=>2 101001100=>2 101001101=>4 101001110=>3 101001111=>3 101010000=>2 101010001=>3 101010010=>2 101010011=>3 101010100=>2 101010101=>1 101010110=>2 101010111=>2 101011000=>3 101011001=>2 101011010=>2 101011011=>3 101011100=>3 101011101=>3 101011110=>2 101011111=>2 101100000=>3 101100001=>2 101100010=>3 101100011=>3 101100100=>3 101100101=>4 101100110=>2 101100111=>3 101101000=>2 101101001=>3 101101010=>2 101101011=>3 101101100=>3 101101101=>1 101101110=>3 101101111=>2 101110000=>3 101110001=>3 101110010=>3 101110011=>2 101110100=>2 101110101=>3 101110110=>3 101110111=>2 101111000=>3 101111001=>3 101111010=>2 101111011=>2 101111100=>3 101111101=>1 101111110=>2 101111111=>2 110000000=>2 110000001=>2 110000010=>3 110000011=>1 110000100=>3 110000101=>3 110000110=>2 110000111=>2 110001000=>2 110001001=>3 110001010=>3 110001011=>4 110001100=>2 110001101=>3 110001110=>3 110001111=>2 110010000=>3 110010001=>3 110010010=>3 110010011=>1 110010100=>2 110010101=>3 110010110=>3 110010111=>4 110011000=>2 110011001=>2 110011010=>2 110011011=>3 110011100=>2 110011101=>2 110011110=>3 110011111=>2 110100000=>3 110100001=>3 110100010=>2 110100011=>4 110100100=>3 110100101=>2 110100110=>3 110100111=>4 110101000=>3 110101001=>3 110101010=>2 110101011=>1 110101100=>2 110101101=>3 110101110=>3 110101111=>2 110110000=>2 110110001=>3 110110010=>3 110110011=>3 110110100=>3 110110101=>3 110110110=>2 110110111=>2 110111000=>3 110111001=>2 110111010=>3 110111011=>1 110111100=>3 110111101=>2 110111110=>2 110111111=>2 111000000=>2 111000001=>2 111000010=>3 111000011=>2 111000100=>3 111000101=>3 111000110=>3 111000111=>1 111001000=>3 111001001=>2 111001010=>3 111001011=>4 111001100=>2 111001101=>3 111001110=>2 111001111=>2 111010000=>3 111010001=>3 111010010=>2 111010011=>4 111010100=>3 111010101=>2 111010110=>3 111010111=>1 111011000=>3 111011001=>3 111011010=>3 111011011=>2 111011100=>2 111011101=>2 111011110=>2 111011111=>2 111100000=>2 111100001=>2 111100010=>3 111100011=>2 111100100=>2 111100101=>3 111100110=>3 111100111=>2 111101000=>3 111101001=>3 111101010=>2 111101011=>2 111101100=>3 111101101=>2 111101110=>2 111101111=>1 111110000=>2 111110001=>2 111110010=>3 111110011=>2 111110100=>3 111110101=>2 111110110=>2 111110111=>2 111111000=>2 111111001=>2 111111010=>2 111111011=>2 111111100=>2 111111101=>2 111111110=>2 111111111=>1 1010101010=>2 1010101100=>3 1010110010=>3 1010110100=>3 1010111000=>3 1011001010=>3 1011001100=>3 1011010010=>3 1011010100=>3 1011011000=>3 1011100010=>4 1011100100=>3 1011101000=>2 1011110000=>3 1100101010=>3 1100101100=>4 1100110010=>3 1100110100=>3 1100111000=>3 1101001010=>3 1101001100=>3 1101010010=>3 1101010100=>3 1101011000=>2 1101100010=>3 1101100100=>2 1101101000=>3 1101110000=>3 1110001010=>3 1110001100=>3 1110010010=>3 1110010100=>2 1110011000=>3 1110100010=>2 1110100100=>3 1110101000=>3 1110110000=>3 1111000010=>3 1111000100=>3 1111001000=>3 1111010000=>3 1111100000=>2 1111111110=>2 1111111101=>2 1111000001=>2 1110111110=>2 1110100001=>3 1110011110=>3 1110010001=>3 1110001001=>3 1110000101=>3 1101111110=>2 1101100001=>3 1101010001=>3 1101001110=>3 1101001001=>3 1101000101=>2 1100110001=>3 1100101001=>2 1100100110=>2 1100100101=>3 1100011001=>3 1100010101=>3 1100010010=>3 1100001000=>3 1011111110=>2 1011111101=>1 1011100001=>3 1011010001=>3 1011001001=>2 1011000101=>4 1010111110=>2 1010110001=>2 1010101001=>3 1010100110=>3 1010100101=>3 1010011010=>4 1010011001=>3 1010010110=>3 1010010101=>3 1010001110=>3 1001111110=>3 1001111000=>3 1001110100=>4 1001110001=>3 1001101100=>2 1001101010=>3 1001101001=>4 1001100110=>3 1001100101=>3 1001011110=>2 1001011010=>3 1001011001=>4 1001010110=>4 1001010101=>3 1001001110=>3 1000111110=>3 1000111100=>3 1000111010=>3 1000110110=>3 1000101110=>2 1000011110=>3 1000010110=>2 1000001110=>3 1000001010=>2 1000000110=>3 1000000100=>2 1000000010=>2 1000000000=>2 0111100001=>3 0000000001=>2 0111010001=>2 0010000001=>2 0101010101=>2 0101010011=>3 0101001101=>3 0101001011=>3 0101000111=>3 0100110101=>3 0100110011=>3 0100101101=>3 0100101011=>3 0100100111=>3 0100011101=>4 0100011011=>3 0100010111=>2 0100001111=>3 0011010101=>3 0011010011=>4 0011001101=>3 0011001011=>3 0011000111=>3 0010110101=>3 0010110011=>3 0010101101=>3 0010101011=>3 0010100111=>2 0010011101=>3 0010011011=>2 0010010111=>3 0010001111=>3 0001110101=>3 0001110011=>3 0001101101=>3 0001101011=>2 0001100111=>3 0001011101=>2 0001011011=>3 0001010111=>3 0001001111=>3 0000111101=>3 0000111011=>3 0000110111=>3 0000101111=>3 0000011111=>2 0000000000=>1 0000000110=>2 0000011000=>2 0000011110=>2 0000010010=>2 0001100000=>2 0001100110=>2 0001111000=>1 0001111110=>2 0001110010=>3 0001001000=>1 0001001110=>3 0001000010=>2 0001011010=>2 0010101010=>2 0010101000=>2 0010101100=>4 0010100100=>3 0010010100=>3 0001010100=>2 0010101110=>3 0010100000=>2 0010111100=>4 0010000100=>1 0011110100=>4 0000010100=>2 0111010100=>3 0000000010=>2 0000001110=>2 0000111110=>2 0000000100=>2 0000001010=>2 0000010110=>3 0000101110=>3 0001011110=>3 0101010110=>3 0101011001=>3 0101011010=>3 0101011100=>3 0101100101=>4 0101100110=>3 0101101001=>3 0101101010=>3 0101101100=>3 0101110001=>3 0101110010=>4 0101110100=>2 0101111000=>3 0110010101=>3 0110010110=>4 0110011001=>3 0110011010=>3 0110011100=>3 0110100101=>3 0110100110=>4 0110101001=>4 0110101010=>3 0110101100=>2 0110110001=>3 0110110010=>2 0110110100=>3 0110111000=>3 0111000101=>3 0111000110=>3 0111001001=>3 0111001010=>2 0111001100=>3 0111010010=>3 0111011000=>3 0111100010=>3 0111100100=>3 0111101000=>3 0111110000=>2 0011100110=>3 0011101010=>3 0011101100=>4 0001101110=>3 0010110110=>3 0011110010=>3 0001110110=>3 0100111010=>4 0010111010=>2 0011111000=>2 0001111010=>3 0100111100=>3 0001111100=>2 0000000101=>2 0000001101=>3 0000011101=>3 0001111101=>3 0000001001=>2 0000010101=>2 0000101101=>2 0010110001=>3 0001100001=>3 0001000001=>2 0000111111=>2 0001110100=>4 0001101100=>2 0010011100=>3 0001011100=>4 0000111100=>2 0001101010=>3 0010011010=>3 0000111010=>3 0010100110=>3 0010010110=>3 0001010110=>3 0000110110=>2 0010001110=>2 0001101001=>3 0000110001=>3 0000010001=>2 0000001100=>2 0000011001=>3 0000110011=>2 0000110100=>4 0000000011=>2 0000011011=>2 0001101111=>3 0000110101=>3 0010000010=>2 0001000011=>3 0010011110=>3 0001010000=>2 0001000100=>2 0001010101=>2 0001001011=>3 0001100011=>2 0001110111=>2 0010000000=>2 0000000111=>2 0000001111=>2 0001111111=>2 0000001011=>3 0000010111=>3 0001011111=>3 0001111001=>3 0000011010=>3 0000101010=>2 0000100101=>3 0010010001=>2 0010000101=>3 0010010101=>2 0010010011=>2 0000101011=>3 0001000111=>2 0000100111=>3 0010010010=>2 0001001010=>3 0001001001=>2 0001000101=>2 0000100011=>3 0000100100=>2 0000100010=>2 0010110000=>4 0000011100=>2 0000001000=>2 0000101100=>4 0010101111=>3 0000101000=>2 0010001000=>2 0001001100=>3 0001000110=>3 0001100100=>3 0001100010=>3 0000100110=>3 0000110010=>3 0010010000=>2 0001010010=>3 0001000000=>2 0001011001=>3 0001100101=>3 0001110001=>2 0010011001=>2 0010100101=>2 0010101001=>2 0010011111=>2 0000111001=>3 0010001101=>3 0001001101=>3 0001010001=>2 0001010011=>3 0001111011=>3 0010011000=>3 0001011000=>4 0000100000=>2 0010001011=>3 0010000111=>3 0000010011=>3 0010001100=>3 0010000110=>3 0000110000=>1 0000100001=>2 0010100011=>3 0010100001=>3 0000111000=>2 0001101000=>4 0000010000=>2 0000101001=>3 0010100010=>2 0010001001=>2 0010000011=>3 0010001010=>2 0001110000=>2 10000000000=>2 10000000010=>2 10000000110=>3 10000001110=>3 10000011110=>3 10000111110=>3 10001111110=>3 10011111110=>3 10111111110=>2 11111111110=>2 10000000100=>2 10000001010=>2 10000010110=>2 10000101110=>2 10001011110=>2 10010111110=>2 10101111110=>2 11011111110=>2 10000010000=>2 10000100010=>3 10001000010=>3 10000100100=>2 10001001010=>3 10001000100=>2 10000101000=>3 10010001010=>3 10001010010=>2 10010000100=>2 10001001000=>2 10010010010=>2 10001010100=>3 10010101010=>2 10010001000=>2 10100010010=>3 10010100010=>2 10010010100=>2 10100101010=>2 00001000001=>2 101010101010=>2 101010101100=>3 101010110010=>3 101010110100=>3 101010111000=>3 101011001010=>3 101011001100=>3 101011010010=>3 101011010100=>2 101011011000=>3 101011100010=>4 101011100100=>3 101011101000=>3 101011110000=>3 101100101010=>3 101100101100=>4 101100110010=>3 101100110100=>2 101100111000=>4 101101001010=>3 101101001100=>2 101101010010=>2 101101010100=>3 101101011000=>3 101101100010=>3 101101100100=>3 101101101000=>2 101101110000=>3 101110001010=>4 101110001100=>3 101110010010=>3 101110010100=>3 101110011000=>3 101110100010=>3 101110100100=>2 101110101000=>3 101110110000=>3 101111000010=>4 101111000100=>3 101111001000=>4 101111010000=>2 101111100000=>3 110010101010=>3 110010101100=>5 110010110010=>4 110010110100=>2 110010111000=>4 110011001010=>3 110011001100=>2 110011010010=>2 110011010100=>3 110011011000=>3 110011100010=>3 110011100100=>3 110011101000=>3 110011110000=>3 110100101010=>3 110100101100=>2 110100110010=>2 110100110100=>5 110100111000=>4 110101001010=>2 110101001100=>3 110101010010=>3 110101010100=>3 110101011000=>2 110101100010=>3 110101100100=>2 110101101000=>3 110101110000=>3 110110001010=>3 110110001100=>3 110110010010=>3 110110010100=>2 110110011000=>3 110110100010=>2 110110100100=>3 110110101000=>3 110110110000=>2 110111000010=>3 110111000100=>3 110111001000=>3 110111010000=>3 110111100000=>3 111000101010=>3 111000101100=>4 111000110010=>4 111000110100=>4 111000111000=>2 111001001010=>3 111001001100=>3 111001010010=>3 111001010100=>2 111001011000=>4 111001100010=>3 111001100100=>3 111001101000=>4 111001110000=>2 111010001010=>3 111010001100=>3 111010010010=>2 111010010100=>3 111010011000=>4 111010100010=>3 111010100100=>3 111010101000=>3 111010110000=>3 111011000010=>4 111011000100=>3 111011001000=>4 111011010000=>3 111011100000=>2 111100001010=>3 111100001100=>3 111100010010=>3 111100010100=>3 111100011000=>2 111100100010=>3 111100100100=>2 111100101000=>3 111100110000=>3 111101000010=>2 111101000100=>3 111101001000=>3 111101010000=>3 111101100000=>3 111110000010=>3 111110000100=>3 111110001000=>2 111110010000=>3 111110100000=>3 111111000000=>2 100001111100=>3 010000011111=>3 000001111101=>3 010111110000=>3 011111000001=>3 000011011110=>3 111100000101=>3 10110111100000=>3 10111011010000=>3 10111011100000=>3 10111100110000=>4 10111101001000=>3 10111101010000=>3 10111101100000=>3 10111110000100=>4 10111110001000=>3 10111110010000=>4 10111110100000=>2 10111111000000=>3 11001111100000=>3 11011110000010=>4 11101101000010=>3 11101110000010=>3 11110011000010=>4 11110100100010=>3 11110101000010=>3 11110110000010=>4 11111000001100=>3 11111000010010=>3 11111000100010=>3 11111001000010=>3 11111010000010=>2 11111100000010=>3 1011110111000000=>3 1011111010100000=>3 1011111011000000=>3 1011111100010000=>4 1011111100100000=>4 1011111101000000=>2 1011111110000000=>3 1111011100000010=>4 1111101010000010=>3 1111101100000010=>4 1111110001000010=>3 1111110010000010=>3 1111110100000010=>2 1111111000000010=>3 111111110000000010=>3 101111111100000000=>3 111111101000000010=>2 101111111010000000=>2 11111111100000000010=>3 10111111111000000000=>3
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
Description
The length of the shortest palindromic decomposition of a binary word.
A palindromic decomposition (paldec for short) of a word $w=a_1,\dots,a_n$ is any list of factors $p_1,\dots,p_k$ such that $w=p_1\dots p_k$ and each $p_i$ is a palindrome, i.e. coincides with itself read backwards.
References
[1] მამუკა ჯიბლაძე Combinatorics of palindromic decompositions MathOverflow:201205
[2] Ravsky, A. On the palindromic decomposition of binary words arXiv:1004.1278
Code
def statistic(w):
    return min(len(p) for p in paldecs(w))

def paldecs(w):
    """
    Return all distinct palindromic decompositions of w.

    sage: w = Word([1,1])
    sage: paldecs(w)
    [[word: 1, word: 1], [word: 11]]

    sage: w = Word([1,0,1])
    sage: paldecs(w)
    [[word: 1, word: 0, word: 1], [word: 101]]

    sage: w = Word([1,0,1,0,0,1])
    sage: paldecs(w)
    [[word: 1, word: 0, word: 1, word: 0, word: 0, word: 1],
     [word: 1, word: 0, word: 1, word: 00, word: 1],
     [word: 1, word: 0, word: 1001],
     [word: 1, word: 010, word: 0, word: 1],
     [word: 101, word: 0, word: 0, word: 1],
     [word: 101, word: 00, word: 1]]

    sage: w = Word("referee")
    sage: paldecs(w)
    [[word: r, word: e, word: f, word: e, word: r, word: e, word: e],
     [word: r, word: e, word: f, word: e, word: r, word: ee],
     [word: r, word: e, word: f, word: ere, word: e],
     [word: r, word: efe, word: r, word: e, word: e],
     [word: r, word: efe, word: r, word: ee],
     [word: refer, word: e, word: e],
     [word: refer, word: ee]]

    """
    if len(w) == 0:
        return [[]]
    P1 = w.palindrome_prefixes()
    result = []
    for p1 in P1:
        l = len(p1)
        if l > 0:
            P2 = paldecs(w[l:])
            for p2 in P2:
                result.append([p1] + p2)
    return result
Created
Oct 16, 2016 at 21:19 by Martin Rubey
Updated
Dec 30, 2017 at 22:39 by Martin Rubey