Identifier
Values
[1] => 1 => 1 => 1
[1,1] => 11 => 11 => 2
[2] => 10 => 11 => 2
[1,1,1] => 111 => 111 => 3
[1,2] => 110 => 111 => 3
[2,1] => 101 => 110 => 2
[3] => 100 => 101 => 2
[1,1,1,1] => 1111 => 1111 => 4
[1,1,2] => 1110 => 1111 => 4
[1,2,1] => 1101 => 1110 => 3
[1,3] => 1100 => 1101 => 3
[2,1,1] => 1011 => 1101 => 3
[2,2] => 1010 => 1101 => 3
[3,1] => 1001 => 1010 => 2
[4] => 1000 => 1001 => 2
[1,1,1,1,1] => 11111 => 11111 => 5
[1,1,1,2] => 11110 => 11111 => 5
[1,1,2,1] => 11101 => 11110 => 4
[1,1,3] => 11100 => 11101 => 4
[1,2,1,1] => 11011 => 11101 => 4
[1,2,2] => 11010 => 11101 => 4
[1,3,1] => 11001 => 11010 => 3
[1,4] => 11000 => 11001 => 3
[2,1,1,1] => 10111 => 11011 => 4
[2,1,2] => 10110 => 11011 => 4
[2,2,1] => 10101 => 11010 => 3
[2,3] => 10100 => 11001 => 3
[3,1,1] => 10011 => 10101 => 3
[3,2] => 10010 => 10101 => 3
[4,1] => 10001 => 10010 => 2
[5] => 10000 => 10001 => 2
[1,1,1,1,1,1] => 111111 => 111111 => 6
[1,1,1,1,2] => 111110 => 111111 => 6
[1,1,1,2,1] => 111101 => 111110 => 5
[1,1,1,3] => 111100 => 111101 => 5
[1,1,2,1,1] => 111011 => 111101 => 5
[1,1,2,2] => 111010 => 111101 => 5
[1,1,3,1] => 111001 => 111010 => 4
[1,1,4] => 111000 => 111001 => 4
[1,2,1,1,1] => 110111 => 111011 => 5
[1,2,1,2] => 110110 => 111011 => 5
[1,2,2,1] => 110101 => 111010 => 4
[1,2,3] => 110100 => 111001 => 4
[1,3,1,1] => 110011 => 110101 => 4
[1,3,2] => 110010 => 110101 => 4
[1,4,1] => 110001 => 110010 => 3
[1,5] => 110000 => 110001 => 3
[2,1,1,1,1] => 101111 => 110111 => 5
[2,1,1,2] => 101110 => 110111 => 5
[2,1,2,1] => 101101 => 110110 => 4
[2,1,3] => 101100 => 110101 => 4
[2,2,1,1] => 101011 => 110101 => 4
[2,2,2] => 101010 => 110101 => 4
[2,3,1] => 101001 => 110010 => 3
[2,4] => 101000 => 110001 => 3
[3,1,1,1] => 100111 => 101011 => 4
[3,1,2] => 100110 => 101011 => 4
[3,2,1] => 100101 => 101010 => 3
[3,3] => 100100 => 101001 => 3
[4,1,1] => 100011 => 100101 => 3
[4,2] => 100010 => 100101 => 3
[5,1] => 100001 => 100010 => 2
[6] => 100000 => 100001 => 2
[1,1,1,1,1,1,1] => 1111111 => 1111111 => 7
[1,1,1,1,1,2] => 1111110 => 1111111 => 7
[1,1,1,1,2,1] => 1111101 => 1111110 => 6
[1,1,1,1,3] => 1111100 => 1111101 => 6
[1,1,1,2,1,1] => 1111011 => 1111101 => 6
[1,1,1,2,2] => 1111010 => 1111101 => 6
[1,1,1,3,1] => 1111001 => 1111010 => 5
[1,1,1,4] => 1111000 => 1111001 => 5
[1,1,2,1,1,1] => 1110111 => 1111011 => 6
[1,1,2,1,2] => 1110110 => 1111011 => 6
[1,1,2,2,1] => 1110101 => 1111010 => 5
[1,1,2,3] => 1110100 => 1111001 => 5
[1,1,3,1,1] => 1110011 => 1110101 => 5
[1,1,3,2] => 1110010 => 1110101 => 5
[1,1,4,1] => 1110001 => 1110010 => 4
[1,1,5] => 1110000 => 1110001 => 4
[1,2,1,1,1,1] => 1101111 => 1110111 => 6
[1,2,1,1,2] => 1101110 => 1110111 => 6
[1,2,1,2,1] => 1101101 => 1110110 => 5
[1,2,1,3] => 1101100 => 1110101 => 5
[1,2,2,1,1] => 1101011 => 1110101 => 5
[1,2,2,2] => 1101010 => 1110101 => 5
[1,2,3,1] => 1101001 => 1110010 => 4
[1,2,4] => 1101000 => 1110001 => 4
[1,3,1,1,1] => 1100111 => 1101011 => 5
[1,3,1,2] => 1100110 => 1101011 => 5
[1,3,2,1] => 1100101 => 1101010 => 4
[1,3,3] => 1100100 => 1101001 => 4
[1,4,1,1] => 1100011 => 1100101 => 4
[1,4,2] => 1100010 => 1100101 => 4
[1,5,1] => 1100001 => 1100010 => 3
[1,6] => 1100000 => 1100001 => 3
[2,1,1,1,1,1] => 1011111 => 1101111 => 6
[2,1,1,1,2] => 1011110 => 1101111 => 6
[2,1,1,2,1] => 1011101 => 1101110 => 5
[2,1,1,3] => 1011100 => 1101101 => 5
[2,1,2,1,1] => 1011011 => 1101101 => 5
[2,1,2,2] => 1011010 => 1101101 => 5
>>> Load all 367 entries. <<<
[2,1,3,1] => 1011001 => 1101010 => 4
[2,1,4] => 1011000 => 1101001 => 4
[2,2,1,1,1] => 1010111 => 1101011 => 5
[2,2,1,2] => 1010110 => 1101011 => 5
[2,2,2,1] => 1010101 => 1101010 => 4
[2,2,3] => 1010100 => 1101001 => 4
[2,3,1,1] => 1010011 => 1100101 => 4
[2,3,2] => 1010010 => 1100101 => 4
[2,4,1] => 1010001 => 1100010 => 3
[2,5] => 1010000 => 1100001 => 3
[3,1,1,1,1] => 1001111 => 1010111 => 5
[3,1,1,2] => 1001110 => 1010111 => 5
[3,1,2,1] => 1001101 => 1010110 => 4
[3,1,3] => 1001100 => 1010101 => 4
[3,2,1,1] => 1001011 => 1010101 => 4
[3,2,2] => 1001010 => 1010101 => 4
[3,3,1] => 1001001 => 1010010 => 3
[3,4] => 1001000 => 1010001 => 3
[4,1,1,1] => 1000111 => 1001011 => 4
[4,1,2] => 1000110 => 1001011 => 4
[4,2,1] => 1000101 => 1001010 => 3
[4,3] => 1000100 => 1001001 => 3
[5,1,1] => 1000011 => 1000101 => 3
[5,2] => 1000010 => 1000101 => 3
[6,1] => 1000001 => 1000010 => 2
[7] => 1000000 => 1000001 => 2
[1,1,1,1,1,1,1,1] => 11111111 => 11111111 => 8
[1,1,1,1,1,1,2] => 11111110 => 11111111 => 8
[1,1,1,1,1,2,1] => 11111101 => 11111110 => 7
[1,1,1,1,1,3] => 11111100 => 11111101 => 7
[1,1,1,1,2,1,1] => 11111011 => 11111101 => 7
[1,1,1,1,2,2] => 11111010 => 11111101 => 7
[1,1,1,1,3,1] => 11111001 => 11111010 => 6
[1,1,1,1,4] => 11111000 => 11111001 => 6
[1,1,1,2,1,1,1] => 11110111 => 11111011 => 7
[1,1,1,2,1,2] => 11110110 => 11111011 => 7
[1,1,1,2,2,1] => 11110101 => 11111010 => 6
[1,1,1,2,3] => 11110100 => 11111001 => 6
[1,1,1,3,1,1] => 11110011 => 11110101 => 6
[1,1,1,3,2] => 11110010 => 11110101 => 6
[1,1,1,4,1] => 11110001 => 11110010 => 5
[1,1,1,5] => 11110000 => 11110001 => 5
[1,1,2,1,1,1,1] => 11101111 => 11110111 => 7
[1,1,2,1,1,2] => 11101110 => 11110111 => 7
[1,1,2,1,2,1] => 11101101 => 11110110 => 6
[1,1,2,1,3] => 11101100 => 11110101 => 6
[1,1,2,2,1,1] => 11101011 => 11110101 => 6
[1,1,2,2,2] => 11101010 => 11110101 => 6
[1,1,2,3,1] => 11101001 => 11110010 => 5
[1,1,2,4] => 11101000 => 11110001 => 5
[1,1,3,1,1,1] => 11100111 => 11101011 => 6
[1,1,3,1,2] => 11100110 => 11101011 => 6
[1,1,3,2,1] => 11100101 => 11101010 => 5
[1,1,3,3] => 11100100 => 11101001 => 5
[1,1,4,1,1] => 11100011 => 11100101 => 5
[1,1,4,2] => 11100010 => 11100101 => 5
[1,1,5,1] => 11100001 => 11100010 => 4
[1,1,6] => 11100000 => 11100001 => 4
[1,2,1,1,1,1,1] => 11011111 => 11101111 => 7
[1,2,1,1,1,2] => 11011110 => 11101111 => 7
[1,2,1,1,2,1] => 11011101 => 11101110 => 6
[1,2,1,1,3] => 11011100 => 11101101 => 6
[1,2,1,2,1,1] => 11011011 => 11101101 => 6
[1,2,1,2,2] => 11011010 => 11101101 => 6
[1,2,1,3,1] => 11011001 => 11101010 => 5
[1,2,1,4] => 11011000 => 11101001 => 5
[1,2,2,1,1,1] => 11010111 => 11101011 => 6
[1,2,2,1,2] => 11010110 => 11101011 => 6
[1,2,2,2,1] => 11010101 => 11101010 => 5
[1,2,2,3] => 11010100 => 11101001 => 5
[1,2,3,1,1] => 11010011 => 11100101 => 5
[1,2,3,2] => 11010010 => 11100101 => 5
[1,2,4,1] => 11010001 => 11100010 => 4
[1,2,5] => 11010000 => 11100001 => 4
[1,3,1,1,1,1] => 11001111 => 11010111 => 6
[1,3,1,1,2] => 11001110 => 11010111 => 6
[1,3,1,2,1] => 11001101 => 11010110 => 5
[1,3,1,3] => 11001100 => 11010101 => 5
[1,3,2,1,1] => 11001011 => 11010101 => 5
[1,3,2,2] => 11001010 => 11010101 => 5
[1,3,3,1] => 11001001 => 11010010 => 4
[1,3,4] => 11001000 => 11010001 => 4
[1,4,1,1,1] => 11000111 => 11001011 => 5
[1,4,1,2] => 11000110 => 11001011 => 5
[1,4,2,1] => 11000101 => 11001010 => 4
[1,4,3] => 11000100 => 11001001 => 4
[1,5,1,1] => 11000011 => 11000101 => 4
[1,5,2] => 11000010 => 11000101 => 4
[1,6,1] => 11000001 => 11000010 => 3
[1,7] => 11000000 => 11000001 => 3
[2,1,1,1,1,1,1] => 10111111 => 11011111 => 7
[2,1,1,1,1,2] => 10111110 => 11011111 => 7
[2,1,1,1,2,1] => 10111101 => 11011110 => 6
[2,1,1,1,3] => 10111100 => 11011101 => 6
[2,1,1,2,1,1] => 10111011 => 11011101 => 6
[2,1,1,2,2] => 10111010 => 11011101 => 6
[2,1,1,3,1] => 10111001 => 11011010 => 5
[2,1,1,4] => 10111000 => 11011001 => 5
[2,1,2,1,1,1] => 10110111 => 11011011 => 6
[2,1,2,1,2] => 10110110 => 11011011 => 6
[2,1,2,2,1] => 10110101 => 11011010 => 5
[2,1,2,3] => 10110100 => 11011001 => 5
[2,1,3,1,1] => 10110011 => 11010101 => 5
[2,1,3,2] => 10110010 => 11010101 => 5
[2,1,4,1] => 10110001 => 11010010 => 4
[2,1,5] => 10110000 => 11010001 => 4
[2,2,1,1,1,1] => 10101111 => 11010111 => 6
[2,2,1,1,2] => 10101110 => 11010111 => 6
[2,2,1,2,1] => 10101101 => 11010110 => 5
[2,2,1,3] => 10101100 => 11010101 => 5
[2,2,2,1,1] => 10101011 => 11010101 => 5
[2,2,2,2] => 10101010 => 11010101 => 5
[2,2,3,1] => 10101001 => 11010010 => 4
[2,2,4] => 10101000 => 11010001 => 4
[2,3,1,1,1] => 10100111 => 11001011 => 5
[2,3,1,2] => 10100110 => 11001011 => 5
[2,3,2,1] => 10100101 => 11001010 => 4
[2,3,3] => 10100100 => 11001001 => 4
[2,4,1,1] => 10100011 => 11000101 => 4
[2,4,2] => 10100010 => 11000101 => 4
[2,5,1] => 10100001 => 11000010 => 3
[2,6] => 10100000 => 11000001 => 3
[3,1,1,1,1,1] => 10011111 => 10101111 => 6
[3,1,1,1,2] => 10011110 => 10101111 => 6
[3,1,1,2,1] => 10011101 => 10101110 => 5
[3,1,1,3] => 10011100 => 10101101 => 5
[3,1,2,1,1] => 10011011 => 10101101 => 5
[3,1,2,2] => 10011010 => 10101101 => 5
[3,1,3,1] => 10011001 => 10101010 => 4
[3,1,4] => 10011000 => 10101001 => 4
[3,2,1,1,1] => 10010111 => 10101011 => 5
[3,2,1,2] => 10010110 => 10101011 => 5
[3,2,2,1] => 10010101 => 10101010 => 4
[3,2,3] => 10010100 => 10101001 => 4
[3,3,1,1] => 10010011 => 10100101 => 4
[3,3,2] => 10010010 => 10100101 => 4
[3,4,1] => 10010001 => 10100010 => 3
[3,5] => 10010000 => 10100001 => 3
[4,1,1,1,1] => 10001111 => 10010111 => 5
[4,1,1,2] => 10001110 => 10010111 => 5
[4,1,2,1] => 10001101 => 10010110 => 4
[4,1,3] => 10001100 => 10010101 => 4
[4,2,1,1] => 10001011 => 10010101 => 4
[4,2,2] => 10001010 => 10010101 => 4
[4,3,1] => 10001001 => 10010010 => 3
[4,4] => 10001000 => 10010001 => 3
[5,1,1,1] => 10000111 => 10001011 => 4
[5,1,2] => 10000110 => 10001011 => 4
[5,2,1] => 10000101 => 10001010 => 3
[5,3] => 10000100 => 10001001 => 3
[6,1,1] => 10000011 => 10000101 => 3
[6,2] => 10000010 => 10000101 => 3
[7,1] => 10000001 => 10000010 => 2
[8] => 10000000 => 10000001 => 2
[1,1,1,1,1,1,1,1,1] => 111111111 => 111111111 => 9
[1,1,1,1,1,1,1,2] => 111111110 => 111111111 => 9
[1,1,1,1,1,1,2,1] => 111111101 => 111111110 => 8
[1,1,1,1,1,1,3] => 111111100 => 111111101 => 8
[1,1,1,1,1,2,1,1] => 111111011 => 111111101 => 8
[1,1,1,1,1,2,2] => 111111010 => 111111101 => 8
[1,1,1,1,1,3,1] => 111111001 => 111111010 => 7
[1,1,1,1,2,2,1] => 111110101 => 111111010 => 7
[1,1,1,3,2,1] => 111100101 => 111101010 => 6
[1,1,2,1,1,2,1] => 111011101 => 111101110 => 7
[1,1,2,1,3,1] => 111011001 => 111101010 => 6
[1,1,2,2,2,1] => 111010101 => 111101010 => 6
[1,1,3,1,2,1] => 111001101 => 111010110 => 6
[1,2,1,1,1,2,1] => 110111101 => 111011110 => 7
[1,2,2,1,2,1] => 110101101 => 111010110 => 6
[1,3,1,3,1] => 110011001 => 110101010 => 5
[1,3,2,2,1] => 110010101 => 110101010 => 5
[1,4,1,2,1] => 110001101 => 110010110 => 5
[1,4,3,1] => 110001001 => 110010010 => 4
[2,1,1,1,1,2,1] => 101111101 => 110111110 => 7
[2,1,1,1,1,3] => 101111100 => 110111101 => 7
[2,1,1,1,2,1,1] => 101111011 => 110111101 => 7
[2,1,1,1,2,2] => 101111010 => 110111101 => 7
[2,1,2,1,2,1] => 101101101 => 110110110 => 6
[2,1,3,2,1] => 101100101 => 110101010 => 5
[2,2,1,3,1] => 101011001 => 110101010 => 5
[2,2,2,2,1] => 101010101 => 110101010 => 5
[2,3,1,2,1] => 101001101 => 110010110 => 5
[2,3,3,1] => 101001001 => 110010010 => 4
[3,1,1,1,2,1] => 100111101 => 101011110 => 6
[3,1,1,3,1] => 100111001 => 101011010 => 5
[3,1,2,2,1] => 100110101 => 101011010 => 5
[3,1,3,1,1] => 100110011 => 101010101 => 5
[3,1,3,2] => 100110010 => 101010101 => 5
[3,1,4,1] => 100110001 => 101010010 => 4
[3,2,1,2,1] => 100101101 => 101010110 => 5
[3,2,1,3] => 100101100 => 101010101 => 5
[3,2,2,1,1] => 100101011 => 101010101 => 5
[3,2,2,2] => 100101010 => 101010101 => 5
[3,2,3,1] => 100101001 => 101010010 => 4
[3,3,2,1] => 100100101 => 101001010 => 4
[3,3,3] => 100100100 => 101001001 => 4
[4,1,1,2,1] => 100011101 => 100101110 => 5
[4,1,3,1] => 100011001 => 100101010 => 4
[4,1,4] => 100011000 => 100101001 => 4
[4,2,2,1] => 100010101 => 100101010 => 4
[4,2,3] => 100010100 => 100101001 => 4
[4,3,1,1] => 100010011 => 100100101 => 4
[4,3,2] => 100010010 => 100100101 => 4
[4,4,1] => 100010001 => 100100010 => 3
[5,1,2,1] => 100001101 => 100010110 => 4
[5,3,1] => 100001001 => 100010010 => 3
[5,4] => 100001000 => 100010001 => 3
[6,2,1] => 100000101 => 100001010 => 3
[8,1] => 100000001 => 100000010 => 2
[1,1,1,1,1,1,1,2,1] => 1111111101 => 1111111110 => 9
[1,1,1,6,1] => 1111000001 => 1111000010 => 5
[1,1,2,5,1] => 1110100001 => 1111000010 => 5
[1,1,3,4,1] => 1110010001 => 1110100010 => 5
[1,1,4,3,1] => 1110001001 => 1110010010 => 5
[1,1,5,2,1] => 1110000101 => 1110001010 => 5
[1,2,1,5,1] => 1101100001 => 1110100010 => 5
[1,2,2,4,1] => 1101010001 => 1110100010 => 5
[1,2,3,3,1] => 1101001001 => 1110010010 => 5
[1,2,4,2,1] => 1101000101 => 1110001010 => 5
[1,2,5,1,1] => 1101000011 => 1110000101 => 5
[1,3,1,4,1] => 1100110001 => 1101010010 => 5
[1,3,2,3,1] => 1100101001 => 1101010010 => 5
[1,3,3,2,1] => 1100100101 => 1101001010 => 5
[1,4,1,3,1] => 1100011001 => 1100101010 => 5
[1,4,2,2,1] => 1100010101 => 1100101010 => 5
[1,4,3,2] => 1100010010 => 1100100101 => 5
[1,5,2,1,1] => 1100001011 => 1100010101 => 5
[2,1,1,1,1,1,2,1] => 1011111101 => 1101111110 => 8
[2,1,1,5,1] => 1011100001 => 1101100010 => 5
[2,1,2,4,1] => 1011010001 => 1101100010 => 5
[2,1,3,3,1] => 1011001001 => 1101010010 => 5
[2,1,4,2,1] => 1011000101 => 1101001010 => 5
[2,2,1,4,1] => 1010110001 => 1101010010 => 5
[2,2,2,3,1] => 1010101001 => 1101010010 => 5
[2,2,3,2,1] => 1010100101 => 1101001010 => 5
[2,3,1,3,1] => 1010011001 => 1100101010 => 5
[2,3,2,2,1] => 1010010101 => 1100101010 => 5
[3,1,1,4,1] => 1001110001 => 1010110010 => 5
[3,1,2,3,1] => 1001101001 => 1010110010 => 5
[3,1,3,2,1] => 1001100101 => 1010101010 => 5
[3,2,1,3,1] => 1001011001 => 1010101010 => 5
[3,2,2,2,1] => 1001010101 => 1010101010 => 5
[4,1,1,1,2,1] => 1000111101 => 1001011110 => 6
[4,2,2,1,1] => 1000101011 => 1001010101 => 5
[6,1,2,1] => 1000001101 => 1000010110 => 4
[9,1] => 1000000001 => 1000000010 => 2
[1,2,2,3,3,1] => 110101001001 => 111010010010 => 6
[1,2,3,3,2,1] => 110100100101 => 111001001010 => 6
[1,2,3,2,3,1] => 110100101001 => 111001010010 => 6
[1,3,3,2,2,1] => 110010010101 => 110100101010 => 6
[1,3,2,3,2,1] => 110010100101 => 110101001010 => 6
[1,3,2,2,3,1] => 110010101001 => 110101010010 => 6
[2,3,2,2,2,1] => 101001010101 => 110010101010 => 6
[2,2,3,2,2,1] => 101010010101 => 110100101010 => 6
[2,2,2,3,2,1] => 101010100101 => 110101001010 => 6
[2,2,2,2,3,1] => 101010101001 => 110101010010 => 6
[3,3,3,2] => 10010010010 => 10100100101 => 5
[3,4,4] => 10010001000 => 10100010001 => 4
[4,4,3] => 10001000100 => 10010001001 => 4
[3,2,2,2,2,1] => 100101010101 => 101010101010 => 6
[2,1,1,1,6,1] => 101111000001 => 110111000010 => 6
[1,2,2,4,2,1] => 110101000101 => 111010001010 => 6
[1,2,4,2,2,1] => 110100010101 => 111000101010 => 6
[1,4,2,2,2,1] => 110001010101 => 110010101010 => 6
[1,1,1,1,1,1,1,1,2,1] => 11111111101 => 11111111110 => 10
[1,2,2,2,4,1] => 110101010001 => 111010100010 => 6
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Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Map
valleys-to-peaks
Description
Return the binary word with every valley replaced by a peak.
A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. This map replaces every valley with a peak.
Map
to binary word
Description
Return the composition as a binary word, treating ones as separators.
Encoding a positive integer $i$ as the word $10\dots 0$ consisting of a one followed by $i-1$ zeros, the binary word of a composition $(i_1,\dots,i_k)$ is the concatenation of of words for $i_1,\dots,i_k$.
The image of this map contains precisely the words which do not begin with a $0$.