Identifier
Values
[1] => 1 => 0 => 0
[1,1] => 11 => 01 => 1
[2] => 10 => 11 => 0
[1,1,1] => 111 => 101 => 1
[1,2] => 110 => 010 => 1
[2,1] => 101 => 001 => 3
[3] => 100 => 110 => 0
[1,1,1,1] => 1111 => 0111 => 6
[1,1,2] => 1110 => 1010 => 1
[1,2,1] => 1101 => 1001 => 3
[1,3] => 1100 => 0100 => 1
[2,1,1] => 1011 => 1111 => 0
[2,2] => 1010 => 0010 => 3
[3,1] => 1001 => 0001 => 6
[4] => 1000 => 1100 => 0
[1,1,1,1,1] => 11111 => 10111 => 6
[1,1,1,2] => 11110 => 01110 => 6
[1,1,2,1] => 11101 => 01101 => 8
[1,1,3] => 11100 => 10100 => 1
[1,2,1,1] => 11011 => 01011 => 11
[1,2,2] => 11010 => 10010 => 3
[1,3,1] => 11001 => 10001 => 6
[1,4] => 11000 => 01000 => 1
[2,1,1,1] => 10111 => 00111 => 15
[2,1,2] => 10110 => 11110 => 0
[2,2,1] => 10101 => 11101 => 1
[2,3] => 10100 => 00100 => 3
[3,1,1] => 10011 => 11011 => 3
[3,2] => 10010 => 00010 => 6
[4,1] => 10001 => 00001 => 10
[5] => 10000 => 11000 => 0
[1,1,1,1,1,1] => 111111 => 011111 => 15
[1,1,1,1,2] => 111110 => 101110 => 6
[1,1,1,2,1] => 111101 => 101101 => 8
[1,1,1,3] => 111100 => 011100 => 6
[1,1,2,1,1] => 111011 => 101011 => 11
[1,1,2,2] => 111010 => 011010 => 8
[1,1,3,1] => 111001 => 011001 => 11
[1,1,4] => 111000 => 101000 => 1
[1,2,1,1,1] => 110111 => 100111 => 15
[1,2,1,2] => 110110 => 010110 => 11
[1,2,2,1] => 110101 => 010101 => 14
[1,2,3] => 110100 => 100100 => 3
[1,3,1,1] => 110011 => 010011 => 18
[1,3,2] => 110010 => 100010 => 6
[1,4,1] => 110001 => 100001 => 10
[1,5] => 110000 => 010000 => 1
[2,1,1,1,1] => 101111 => 111111 => 0
[2,1,1,2] => 101110 => 001110 => 15
[2,1,2,1] => 101101 => 001101 => 18
[2,1,3] => 101100 => 111100 => 0
[2,2,1,1] => 101011 => 001011 => 22
[2,2,2] => 101010 => 111010 => 1
[2,3,1] => 101001 => 111001 => 3
[2,4] => 101000 => 001000 => 3
[3,1,1,1] => 100111 => 000111 => 27
[3,1,2] => 100110 => 110110 => 3
[3,2,1] => 100101 => 110101 => 5
[3,3] => 100100 => 000100 => 6
[4,1,1] => 100011 => 110011 => 8
[4,2] => 100010 => 000010 => 10
[5,1] => 100001 => 000001 => 15
[6] => 100000 => 110000 => 0
[1,1,1,1,1,1,1] => 1111111 => 1011111 => 15
[1,1,1,1,1,2] => 1111110 => 0111110 => 15
[1,1,1,1,2,1] => 1111101 => 0111101 => 17
[1,1,1,1,3] => 1111100 => 1011100 => 6
[1,1,1,2,1,1] => 1111011 => 0111011 => 20
[1,1,1,2,2] => 1111010 => 1011010 => 8
[1,1,1,3,1] => 1111001 => 1011001 => 11
[1,1,1,4] => 1111000 => 0111000 => 6
[1,1,2,1,1,1] => 1110111 => 0110111 => 24
[1,1,2,1,2] => 1110110 => 1010110 => 11
[1,1,2,2,1] => 1110101 => 1010101 => 14
[1,1,2,3] => 1110100 => 0110100 => 8
[1,1,3,1,1] => 1110011 => 1010011 => 18
[1,1,3,2] => 1110010 => 0110010 => 11
[1,1,4,1] => 1110001 => 0110001 => 15
[1,1,5] => 1110000 => 1010000 => 1
[1,2,1,1,1,1] => 1101111 => 0101111 => 29
[1,2,1,1,2] => 1101110 => 1001110 => 15
[1,2,1,2,1] => 1101101 => 1001101 => 18
[1,2,1,3] => 1101100 => 0101100 => 11
[1,2,2,1,1] => 1101011 => 1001011 => 22
[1,2,2,2] => 1101010 => 0101010 => 14
[1,2,3,1] => 1101001 => 0101001 => 18
[1,2,4] => 1101000 => 1001000 => 3
[1,3,1,1,1] => 1100111 => 1000111 => 27
[1,3,1,2] => 1100110 => 0100110 => 18
[1,3,2,1] => 1100101 => 0100101 => 22
[1,3,3] => 1100100 => 1000100 => 6
[1,4,1,1] => 1100011 => 0100011 => 27
[1,4,2] => 1100010 => 1000010 => 10
[1,5,1] => 1100001 => 1000001 => 15
[1,6] => 1100000 => 0100000 => 1
[2,1,1,1,1,1] => 1011111 => 0011111 => 35
[2,1,1,1,2] => 1011110 => 1111110 => 0
[2,1,1,2,1] => 1011101 => 1111101 => 1
[2,1,1,3] => 1011100 => 0011100 => 15
[2,1,2,1,1] => 1011011 => 1111011 => 3
[2,1,2,2] => 1011010 => 0011010 => 18
>>> Load all 536 entries. <<<
[2,1,3,1] => 1011001 => 0011001 => 22
[2,1,4] => 1011000 => 1111000 => 0
[2,2,1,1,1] => 1010111 => 1110111 => 6
[2,2,1,2] => 1010110 => 0010110 => 22
[2,2,2,1] => 1010101 => 0010101 => 26
[2,2,3] => 1010100 => 1110100 => 1
[2,3,1,1] => 1010011 => 0010011 => 31
[2,3,2] => 1010010 => 1110010 => 3
[2,4,1] => 1010001 => 1110001 => 6
[2,5] => 1010000 => 0010000 => 3
[3,1,1,1,1] => 1001111 => 1101111 => 10
[3,1,1,2] => 1001110 => 0001110 => 27
[3,1,2,1] => 1001101 => 0001101 => 31
[3,1,3] => 1001100 => 1101100 => 3
[3,2,1,1] => 1001011 => 0001011 => 36
[3,2,2] => 1001010 => 1101010 => 5
[3,3,1] => 1001001 => 1101001 => 8
[3,4] => 1001000 => 0001000 => 6
[4,1,1,1] => 1000111 => 0000111 => 42
[4,1,2] => 1000110 => 1100110 => 8
[4,2,1] => 1000101 => 1100101 => 11
[4,3] => 1000100 => 0000100 => 10
[5,1,1] => 1000011 => 1100011 => 15
[5,2] => 1000010 => 0000010 => 15
[6,1] => 1000001 => 0000001 => 21
[7] => 1000000 => 1100000 => 0
[1,1,1,1,1,1,1,1] => 11111111 => 01111111 => 28
[1,1,1,1,1,1,2] => 11111110 => 10111110 => 15
[1,1,1,1,1,2,1] => 11111101 => 10111101 => 17
[1,1,1,1,1,3] => 11111100 => 01111100 => 15
[1,1,1,1,2,1,1] => 11111011 => 10111011 => 20
[1,1,1,1,2,2] => 11111010 => 01111010 => 17
[1,1,1,1,3,1] => 11111001 => 01111001 => 20
[1,1,1,1,4] => 11111000 => 10111000 => 6
[1,1,1,2,1,1,1] => 11110111 => 10110111 => 24
[1,1,1,2,1,2] => 11110110 => 01110110 => 20
[1,1,1,2,2,1] => 11110101 => 01110101 => 23
[1,1,1,2,3] => 11110100 => 10110100 => 8
[1,1,1,3,1,1] => 11110011 => 01110011 => 27
[1,1,1,3,2] => 11110010 => 10110010 => 11
[1,1,1,4,1] => 11110001 => 10110001 => 15
[1,1,1,5] => 11110000 => 01110000 => 6
[1,1,2,1,1,1,1] => 11101111 => 10101111 => 29
[1,1,2,1,1,2] => 11101110 => 01101110 => 24
[1,1,2,1,2,1] => 11101101 => 01101101 => 27
[1,1,2,1,3] => 11101100 => 10101100 => 11
[1,1,2,2,1,1] => 11101011 => 01101011 => 31
[1,1,2,2,2] => 11101010 => 10101010 => 14
[1,1,2,3,1] => 11101001 => 10101001 => 18
[1,1,2,4] => 11101000 => 01101000 => 8
[1,1,3,1,1,1] => 11100111 => 01100111 => 36
[1,1,3,1,2] => 11100110 => 10100110 => 18
[1,1,3,2,1] => 11100101 => 10100101 => 22
[1,1,3,3] => 11100100 => 01100100 => 11
[1,1,4,1,1] => 11100011 => 10100011 => 27
[1,1,4,2] => 11100010 => 01100010 => 15
[1,1,5,1] => 11100001 => 01100001 => 20
[1,1,6] => 11100000 => 10100000 => 1
[1,2,1,1,1,1,1] => 11011111 => 10011111 => 35
[1,2,1,1,1,2] => 11011110 => 01011110 => 29
[1,2,1,1,2,1] => 11011101 => 01011101 => 32
[1,2,1,1,3] => 11011100 => 10011100 => 15
[1,2,1,2,1,1] => 11011011 => 01011011 => 36
[1,2,1,2,2] => 11011010 => 10011010 => 18
[1,2,1,3,1] => 11011001 => 10011001 => 22
[1,2,1,4] => 11011000 => 01011000 => 11
[1,2,2,1,1,1] => 11010111 => 01010111 => 41
[1,2,2,1,2] => 11010110 => 10010110 => 22
[1,2,2,2,1] => 11010101 => 10010101 => 26
[1,2,2,3] => 11010100 => 01010100 => 14
[1,2,3,1,1] => 11010011 => 10010011 => 31
[1,2,3,2] => 11010010 => 01010010 => 18
[1,2,4,1] => 11010001 => 01010001 => 23
[1,2,5] => 11010000 => 10010000 => 3
[1,3,1,1,1,1] => 11001111 => 01001111 => 47
[1,3,1,1,2] => 11001110 => 10001110 => 27
[1,3,1,2,1] => 11001101 => 10001101 => 31
[1,3,1,3] => 11001100 => 01001100 => 18
[1,3,2,1,1] => 11001011 => 10001011 => 36
[1,3,2,2] => 11001010 => 01001010 => 22
[1,3,3,1] => 11001001 => 01001001 => 27
[1,3,4] => 11001000 => 10001000 => 6
[1,4,1,1,1] => 11000111 => 10000111 => 42
[1,4,1,2] => 11000110 => 01000110 => 27
[1,4,2,1] => 11000101 => 01000101 => 32
[1,4,3] => 11000100 => 10000100 => 10
[1,5,1,1] => 11000011 => 01000011 => 38
[1,5,2] => 11000010 => 10000010 => 15
[1,6,1] => 11000001 => 10000001 => 21
[1,7] => 11000000 => 01000000 => 1
[2,1,1,1,1,1,1] => 10111111 => 11111111 => 0
[2,1,1,1,1,2] => 10111110 => 00111110 => 35
[2,1,1,1,2,1] => 10111101 => 00111101 => 38
[2,1,1,1,3] => 10111100 => 11111100 => 0
[2,1,1,2,1,1] => 10111011 => 00111011 => 42
[2,1,1,2,2] => 10111010 => 11111010 => 1
[2,1,1,3,1] => 10111001 => 11111001 => 3
[2,1,1,4] => 10111000 => 00111000 => 15
[2,1,2,1,1,1] => 10110111 => 00110111 => 47
[2,1,2,1,2] => 10110110 => 11110110 => 3
[2,1,2,2,1] => 10110101 => 11110101 => 5
[2,1,2,3] => 10110100 => 00110100 => 18
[2,1,3,1,1] => 10110011 => 11110011 => 8
[2,1,3,2] => 10110010 => 00110010 => 22
[2,1,4,1] => 10110001 => 00110001 => 27
[2,1,5] => 10110000 => 11110000 => 0
[2,2,1,1,1,1] => 10101111 => 00101111 => 53
[2,2,1,1,2] => 10101110 => 11101110 => 6
[2,2,1,2,1] => 10101101 => 11101101 => 8
[2,2,1,3] => 10101100 => 00101100 => 22
[2,2,2,1,1] => 10101011 => 11101011 => 11
[2,2,2,2] => 10101010 => 00101010 => 26
[2,2,3,1] => 10101001 => 00101001 => 31
[2,2,4] => 10101000 => 11101000 => 1
[2,3,1,1,1] => 10100111 => 11100111 => 15
[2,3,1,2] => 10100110 => 00100110 => 31
[2,3,2,1] => 10100101 => 00100101 => 36
[2,3,3] => 10100100 => 11100100 => 3
[2,4,1,1] => 10100011 => 00100011 => 42
[2,4,2] => 10100010 => 11100010 => 6
[2,5,1] => 10100001 => 11100001 => 10
[2,6] => 10100000 => 00100000 => 3
[3,1,1,1,1,1] => 10011111 => 00011111 => 60
[3,1,1,1,2] => 10011110 => 11011110 => 10
[3,1,1,2,1] => 10011101 => 11011101 => 12
[3,1,1,3] => 10011100 => 00011100 => 27
[3,1,2,1,1] => 10011011 => 11011011 => 15
[3,1,2,2] => 10011010 => 00011010 => 31
[3,1,3,1] => 10011001 => 00011001 => 36
[3,1,4] => 10011000 => 11011000 => 3
[3,2,1,1,1] => 10010111 => 11010111 => 19
[3,2,1,2] => 10010110 => 00010110 => 36
[3,2,2,1] => 10010101 => 00010101 => 41
[3,2,3] => 10010100 => 11010100 => 5
[3,3,1,1] => 10010011 => 00010011 => 47
[3,3,2] => 10010010 => 11010010 => 8
[3,4,1] => 10010001 => 11010001 => 12
[3,5] => 10010000 => 00010000 => 6
[4,1,1,1,1] => 10001111 => 11001111 => 24
[4,1,1,2] => 10001110 => 00001110 => 42
[4,1,2,1] => 10001101 => 00001101 => 47
[4,1,3] => 10001100 => 11001100 => 8
[4,2,1,1] => 10001011 => 00001011 => 53
[4,2,2] => 10001010 => 11001010 => 11
[4,3,1] => 10001001 => 11001001 => 15
[4,4] => 10001000 => 00001000 => 10
[5,1,1,1] => 10000111 => 00000111 => 60
[5,1,2] => 10000110 => 11000110 => 15
[5,2,1] => 10000101 => 11000101 => 19
[5,3] => 10000100 => 00000100 => 15
[6,1,1] => 10000011 => 11000011 => 24
[6,2] => 10000010 => 00000010 => 21
[7,1] => 10000001 => 00000001 => 28
[8] => 10000000 => 11000000 => 0
[1,1,1,1,1,1,1,1,1] => 111111111 => 101111111 => 28
[1,1,1,1,1,1,1,2] => 111111110 => 011111110 => 28
[1,1,1,1,1,1,2,1] => 111111101 => 011111101 => 30
[1,1,1,1,1,1,3] => 111111100 => 101111100 => 15
[1,1,1,1,1,2,1,1] => 111111011 => 011111011 => 33
[1,1,1,1,1,2,2] => 111111010 => 101111010 => 17
[1,1,1,1,1,3,1] => 111111001 => 101111001 => 20
[1,1,1,1,1,4] => 111111000 => 011111000 => 15
[1,1,1,1,2,1,1,1] => 111110111 => 011110111 => 37
[1,1,1,1,2,1,2] => 111110110 => 101110110 => 20
[1,1,1,1,2,2,1] => 111110101 => 101110101 => 23
[1,1,1,1,2,3] => 111110100 => 011110100 => 17
[1,1,1,1,3,1,1] => 111110011 => 101110011 => 27
[1,1,1,1,3,2] => 111110010 => 011110010 => 20
[1,1,1,1,4,1] => 111110001 => 011110001 => 24
[1,1,1,1,5] => 111110000 => 101110000 => 6
[1,1,1,2,1,1,1,1] => 111101111 => 011101111 => 42
[1,1,1,2,1,1,2] => 111101110 => 101101110 => 24
[1,1,1,2,1,2,1] => 111101101 => 101101101 => 27
[1,1,1,2,1,3] => 111101100 => 011101100 => 20
[1,1,1,2,2,1,1] => 111101011 => 101101011 => 31
[1,1,1,2,2,2] => 111101010 => 011101010 => 23
[1,1,1,2,3,1] => 111101001 => 011101001 => 27
[1,1,1,2,4] => 111101000 => 101101000 => 8
[1,1,1,3,1,1,1] => 111100111 => 101100111 => 36
[1,1,1,3,1,2] => 111100110 => 011100110 => 27
[1,1,1,3,2,1] => 111100101 => 011100101 => 31
[1,1,1,3,3] => 111100100 => 101100100 => 11
[1,1,1,4,1,1] => 111100011 => 011100011 => 36
[1,1,1,4,2] => 111100010 => 101100010 => 15
[1,1,1,5,1] => 111100001 => 101100001 => 20
[1,1,1,6] => 111100000 => 011100000 => 6
[1,1,2,1,1,1,1,1] => 111011111 => 011011111 => 48
[1,1,2,1,1,1,2] => 111011110 => 101011110 => 29
[1,1,2,1,1,2,1] => 111011101 => 101011101 => 32
[1,1,2,1,1,3] => 111011100 => 011011100 => 24
[1,1,2,1,2,1,1] => 111011011 => 101011011 => 36
[1,1,2,1,2,2] => 111011010 => 011011010 => 27
[1,1,2,1,3,1] => 111011001 => 011011001 => 31
[1,1,2,1,4] => 111011000 => 101011000 => 11
[1,1,2,2,1,1,1] => 111010111 => 101010111 => 41
[1,1,2,2,1,2] => 111010110 => 011010110 => 31
[1,1,2,2,2,1] => 111010101 => 011010101 => 35
[1,1,2,2,3] => 111010100 => 101010100 => 14
[1,1,2,3,1,1] => 111010011 => 011010011 => 40
[1,1,2,3,2] => 111010010 => 101010010 => 18
[1,1,2,4,1] => 111010001 => 101010001 => 23
[1,1,2,5] => 111010000 => 011010000 => 8
[1,1,3,1,1,1,1] => 111001111 => 101001111 => 47
[1,1,3,1,1,2] => 111001110 => 011001110 => 36
[1,1,3,1,2,1] => 111001101 => 011001101 => 40
[1,1,3,1,3] => 111001100 => 101001100 => 18
[1,1,3,2,1,1] => 111001011 => 011001011 => 45
[1,1,3,2,2] => 111001010 => 101001010 => 22
[1,1,3,3,1] => 111001001 => 101001001 => 27
[1,1,3,4] => 111001000 => 011001000 => 11
[1,1,4,1,1,1] => 111000111 => 011000111 => 51
[1,1,4,1,2] => 111000110 => 101000110 => 27
[1,1,4,2,1] => 111000101 => 101000101 => 32
[1,1,4,3] => 111000100 => 011000100 => 15
[1,1,5,1,1] => 111000011 => 101000011 => 38
[1,1,5,2] => 111000010 => 011000010 => 20
[1,1,6,1] => 111000001 => 011000001 => 26
[1,1,7] => 111000000 => 101000000 => 1
[1,2,1,1,1,1,1,1] => 110111111 => 010111111 => 55
[1,2,1,1,1,1,2] => 110111110 => 100111110 => 35
[1,2,1,1,1,2,1] => 110111101 => 100111101 => 38
[1,2,1,1,1,3] => 110111100 => 010111100 => 29
[1,2,1,1,2,1,1] => 110111011 => 100111011 => 42
[1,2,1,1,2,2] => 110111010 => 010111010 => 32
[1,2,1,1,3,1] => 110111001 => 010111001 => 36
[1,2,1,1,4] => 110111000 => 100111000 => 15
[1,2,1,2,1,1,1] => 110110111 => 100110111 => 47
[1,2,1,2,1,2] => 110110110 => 010110110 => 36
[1,2,1,2,2,1] => 110110101 => 010110101 => 40
[1,2,1,2,3] => 110110100 => 100110100 => 18
[1,2,1,3,1,1] => 110110011 => 010110011 => 45
[1,2,1,3,2] => 110110010 => 100110010 => 22
[1,2,1,4,1] => 110110001 => 100110001 => 27
[1,2,1,5] => 110110000 => 010110000 => 11
[1,2,2,1,1,1,1] => 110101111 => 100101111 => 53
[1,2,2,1,1,2] => 110101110 => 010101110 => 41
[1,2,2,1,2,1] => 110101101 => 010101101 => 45
[1,2,2,1,3] => 110101100 => 100101100 => 22
[1,2,2,2,1,1] => 110101011 => 010101011 => 50
[1,2,2,2,2] => 110101010 => 100101010 => 26
[1,2,2,3,1] => 110101001 => 100101001 => 31
[1,2,2,4] => 110101000 => 010101000 => 14
[1,2,3,1,1,1] => 110100111 => 010100111 => 56
[1,2,3,1,2] => 110100110 => 100100110 => 31
[1,2,3,2,1] => 110100101 => 100100101 => 36
[1,2,3,3] => 110100100 => 010100100 => 18
[1,2,4,1,1] => 110100011 => 100100011 => 42
[1,2,4,2] => 110100010 => 010100010 => 23
[1,2,5,1] => 110100001 => 010100001 => 29
[1,2,6] => 110100000 => 100100000 => 3
[1,3,1,1,1,1,1] => 110011111 => 100011111 => 60
[1,3,1,1,1,2] => 110011110 => 010011110 => 47
[1,3,1,1,2,1] => 110011101 => 010011101 => 51
[1,3,1,1,3] => 110011100 => 100011100 => 27
[1,3,1,2,1,1] => 110011011 => 010011011 => 56
[1,3,1,2,2] => 110011010 => 100011010 => 31
[1,3,1,3,1] => 110011001 => 100011001 => 36
[1,3,1,4] => 110011000 => 010011000 => 18
[1,3,2,1,1,1] => 110010111 => 010010111 => 62
[1,3,2,1,2] => 110010110 => 100010110 => 36
[1,3,2,2,1] => 110010101 => 100010101 => 41
[1,3,2,3] => 110010100 => 010010100 => 22
[1,3,3,1,1] => 110010011 => 100010011 => 47
[1,3,3,2] => 110010010 => 010010010 => 27
[1,3,4,1] => 110010001 => 010010001 => 33
[1,3,5] => 110010000 => 100010000 => 6
[1,4,1,1,1,1] => 110001111 => 010001111 => 69
[1,4,1,1,2] => 110001110 => 100001110 => 42
[1,4,1,2,1] => 110001101 => 100001101 => 47
[1,4,1,3] => 110001100 => 010001100 => 27
[1,4,2,1,1] => 110001011 => 100001011 => 53
[1,4,2,2] => 110001010 => 010001010 => 32
[1,4,3,1] => 110001001 => 010001001 => 38
[1,4,4] => 110001000 => 100001000 => 10
[1,5,1,1,1] => 110000111 => 100000111 => 60
[1,5,1,2] => 110000110 => 010000110 => 38
[1,5,2,1] => 110000101 => 010000101 => 44
[1,5,3] => 110000100 => 100000100 => 15
[1,6,1,1] => 110000011 => 010000011 => 51
[1,6,2] => 110000010 => 100000010 => 21
[1,7,1] => 110000001 => 100000001 => 28
[1,8] => 110000000 => 010000000 => 1
[2,1,1,1,1,1,1,1] => 101111111 => 001111111 => 63
[2,1,1,1,1,1,2] => 101111110 => 111111110 => 0
[2,1,1,1,1,2,1] => 101111101 => 111111101 => 1
[2,1,1,1,1,3] => 101111100 => 001111100 => 35
[2,1,1,1,2,1,1] => 101111011 => 111111011 => 3
[2,1,1,1,2,2] => 101111010 => 001111010 => 38
[2,1,1,1,3,1] => 101111001 => 001111001 => 42
[2,1,1,1,4] => 101111000 => 111111000 => 0
[2,1,1,2,1,1,1] => 101110111 => 111110111 => 6
[2,1,1,2,1,2] => 101110110 => 001110110 => 42
[2,1,1,2,2,1] => 101110101 => 001110101 => 46
[2,1,1,2,3] => 101110100 => 111110100 => 1
[2,1,1,3,1,1] => 101110011 => 001110011 => 51
[2,1,1,3,2] => 101110010 => 111110010 => 3
[2,1,1,4,1] => 101110001 => 111110001 => 6
[2,1,1,5] => 101110000 => 001110000 => 15
[2,1,2,1,1,1,1] => 101101111 => 111101111 => 10
[2,1,2,1,1,2] => 101101110 => 001101110 => 47
[2,1,2,1,2,1] => 101101101 => 001101101 => 51
[2,1,2,1,3] => 101101100 => 111101100 => 3
[2,1,2,2,1,1] => 101101011 => 001101011 => 56
[2,1,2,2,2] => 101101010 => 111101010 => 5
[2,1,2,3,1] => 101101001 => 111101001 => 8
[2,1,2,4] => 101101000 => 001101000 => 18
[2,1,3,1,1,1] => 101100111 => 001100111 => 62
[2,1,3,1,2] => 101100110 => 111100110 => 8
[2,1,3,2,1] => 101100101 => 111100101 => 11
[2,1,3,3] => 101100100 => 001100100 => 22
[2,1,4,1,1] => 101100011 => 111100011 => 15
[2,1,4,2] => 101100010 => 001100010 => 27
[2,1,5,1] => 101100001 => 001100001 => 33
[2,1,6] => 101100000 => 111100000 => 0
[2,2,1,1,1,1,1] => 101011111 => 111011111 => 15
[2,2,1,1,1,2] => 101011110 => 001011110 => 53
[2,2,1,1,2,1] => 101011101 => 001011101 => 57
[2,2,1,1,3] => 101011100 => 111011100 => 6
[2,2,1,2,1,1] => 101011011 => 001011011 => 62
[2,2,1,2,2] => 101011010 => 111011010 => 8
[2,2,1,3,1] => 101011001 => 111011001 => 11
[2,2,1,4] => 101011000 => 001011000 => 22
[2,2,2,1,1,1] => 101010111 => 001010111 => 68
[2,2,2,1,2] => 101010110 => 111010110 => 11
[2,2,2,2,1] => 101010101 => 111010101 => 14
[2,2,2,3] => 101010100 => 001010100 => 26
[2,2,3,1,1] => 101010011 => 111010011 => 18
[2,2,3,2] => 101010010 => 001010010 => 31
[2,2,4,1] => 101010001 => 001010001 => 37
[2,2,5] => 101010000 => 111010000 => 1
[2,3,1,1,1,1] => 101001111 => 001001111 => 75
[2,3,1,1,2] => 101001110 => 111001110 => 15
[2,3,1,2,1] => 101001101 => 111001101 => 18
[2,3,1,3] => 101001100 => 001001100 => 31
[2,3,2,1,1] => 101001011 => 111001011 => 22
[2,3,2,2] => 101001010 => 001001010 => 36
[2,3,3,1] => 101001001 => 001001001 => 42
[2,3,4] => 101001000 => 111001000 => 3
[2,4,1,1,1] => 101000111 => 111000111 => 27
[2,4,1,2] => 101000110 => 001000110 => 42
[2,4,2,1] => 101000101 => 001000101 => 48
[2,4,3] => 101000100 => 111000100 => 6
[2,5,1,1] => 101000011 => 001000011 => 55
[2,5,2] => 101000010 => 111000010 => 10
[2,6,1] => 101000001 => 111000001 => 15
[2,7] => 101000000 => 001000000 => 3
[3,1,1,1,1,1,1] => 100111111 => 110111111 => 21
[3,1,1,1,1,2] => 100111110 => 000111110 => 60
[3,1,1,1,2,1] => 100111101 => 000111101 => 64
[3,1,1,1,3] => 100111100 => 110111100 => 10
[3,1,1,2,1,1] => 100111011 => 000111011 => 69
[3,1,1,2,2] => 100111010 => 110111010 => 12
[3,1,1,3,1] => 100111001 => 110111001 => 15
[3,1,1,4] => 100111000 => 000111000 => 27
[3,1,2,1,1,1] => 100110111 => 000110111 => 75
[3,1,2,1,2] => 100110110 => 110110110 => 15
[3,1,2,2,1] => 100110101 => 110110101 => 18
[3,1,2,3] => 100110100 => 000110100 => 31
[3,1,3,1,1] => 100110011 => 110110011 => 22
[3,1,3,2] => 100110010 => 000110010 => 36
[3,1,4,1] => 100110001 => 000110001 => 42
[3,1,5] => 100110000 => 110110000 => 3
[3,2,1,1,1,1] => 100101111 => 000101111 => 82
[3,2,1,1,2] => 100101110 => 110101110 => 19
[3,2,1,2,1] => 100101101 => 110101101 => 22
[3,2,1,3] => 100101100 => 000101100 => 36
[3,2,2,1,1] => 100101011 => 110101011 => 26
[3,2,2,2] => 100101010 => 000101010 => 41
[3,2,3,1] => 100101001 => 000101001 => 47
[3,2,4] => 100101000 => 110101000 => 5
[3,3,1,1,1] => 100100111 => 110100111 => 31
[3,3,1,2] => 100100110 => 000100110 => 47
[3,3,2,1] => 100100101 => 000100101 => 53
[3,3,3] => 100100100 => 110100100 => 8
[3,4,1,1] => 100100011 => 000100011 => 60
[3,4,2] => 100100010 => 110100010 => 12
[3,5,1] => 100100001 => 110100001 => 17
[3,6] => 100100000 => 000100000 => 6
[4,1,1,1,1,1] => 100011111 => 000011111 => 90
[4,1,1,1,2] => 100011110 => 110011110 => 24
[4,1,1,2,1] => 100011101 => 110011101 => 27
[4,1,1,3] => 100011100 => 000011100 => 42
[4,1,2,1,1] => 100011011 => 110011011 => 31
[4,1,2,2] => 100011010 => 000011010 => 47
[4,1,3,1] => 100011001 => 000011001 => 53
[4,1,4] => 100011000 => 110011000 => 8
[4,2,1,1,1] => 100010111 => 110010111 => 36
[4,2,1,2] => 100010110 => 000010110 => 53
[4,2,2,1] => 100010101 => 000010101 => 59
[4,2,3] => 100010100 => 110010100 => 11
[4,3,1,1] => 100010011 => 000010011 => 66
[4,3,2] => 100010010 => 110010010 => 15
[4,4,1] => 100010001 => 110010001 => 20
[4,5] => 100010000 => 000010000 => 10
[5,1,1,1,1] => 100001111 => 110001111 => 42
[5,1,1,2] => 100001110 => 000001110 => 60
[5,1,2,1] => 100001101 => 000001101 => 66
[5,1,3] => 100001100 => 110001100 => 15
[5,2,1,1] => 100001011 => 000001011 => 73
[5,2,2] => 100001010 => 110001010 => 19
[5,3,1] => 100001001 => 110001001 => 24
[5,4] => 100001000 => 000001000 => 15
[6,1,1,1] => 100000111 => 000000111 => 81
[6,1,2] => 100000110 => 110000110 => 24
[6,2,1] => 100000101 => 110000101 => 29
[6,3] => 100000100 => 000000100 => 21
[7,1,1] => 100000011 => 110000011 => 35
[7,2] => 100000010 => 000000010 => 28
[8,1] => 100000001 => 000000001 => 36
[9] => 100000000 => 110000000 => 0
[2,2,2,2,1,1] => 1010101011 => 0010101011 => 79
[2,3,1,1,2,1] => 1010011101 => 0010011101 => 80
[2,3,2,1,1,1] => 1010010111 => 0010010111 => 93
[2,3,4,1] => 1010010001 => 0010010001 => 49
[2,5,2,1] => 1010000101 => 0010000101 => 62
[3,1,1,1,1,1,1,1] => 1001111111 => 0001111111 => 105
[3,1,1,1,3,1] => 1001111001 => 0001111001 => 69
[3,1,2,1,2,1] => 1001101101 => 0001101101 => 80
[3,1,3,1,1,1] => 1001100111 => 0001100111 => 93
[3,1,5,1] => 1001100001 => 0001100001 => 49
[3,2,1,1,1,2] => 1001011110 => 0001011110 => 82
[4,1,1,1,1,2] => 1000111110 => 0000111110 => 90
[4,1,1,1,2,1] => 1000111101 => 0000111101 => 95
[4,1,2,1,1,1] => 1000110111 => 0000110111 => 108
[4,1,4,1] => 1000110001 => 0000110001 => 60
[4,3,2,1] => 1000100101 => 0000100101 => 73
[5,1,1,1,1,1] => 1000011111 => 0000011111 => 125
[5,1,3,1] => 1000011001 => 0000011001 => 73
[5,2,1,2] => 1000010110 => 0000010110 => 73
[6,1,1,2] => 1000001110 => 0000001110 => 81
[6,1,2,1] => 1000001101 => 0000001101 => 88
[7,1,1,1] => 1000000111 => 0000000111 => 105
[7,3] => 1000000100 => 0000000100 => 28
[8,2] => 1000000010 => 0000000010 => 36
[9,1] => 1000000001 => 0000000001 => 45
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Description
The non-inversion sum of a binary word.
A pair $a < b$ is an noninversion of a binary word $w = w_1 \cdots w_n$ if $w_a < w_b$. The non-inversion sum is given by $\sum(b-a)$ over all non-inversions of $w$.
Map
Gray next
Description
The next element in the Gray code for binary words.
Let $w$ be a binary word. If the number of ones in $w$ is even, return the binary word obtained by flipping the first bit of $w$. If the only one in $w$ is the final bit, return the zero word. Otherwise, return the word obtained by flipping the bit after the first one.
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 $0$.