Identifier
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
Images
[1] => [1] => 1 => 1
[1,1] => [1,1] => 11 => 11
[2] => [2] => 10 => 00
[1,1,1] => [1,1,1] => 111 => 111
[1,2] => [2,1] => 101 => 100
[2,1] => [1,2] => 110 => 001
[3] => [3] => 100 => 010
[1,1,1,1] => [1,1,1,1] => 1111 => 1111
[1,1,2] => [2,1,1] => 1011 => 1100
[1,2,1] => [1,2,1] => 1101 => 1001
[1,3] => [3,1] => 1001 => 1010
[2,1,1] => [1,1,2] => 1110 => 0011
[2,2] => [2,2] => 1010 => 0000
[3,1] => [1,3] => 1100 => 0101
[4] => [4] => 1000 => 0110
[1,1,1,1,1] => [1,1,1,1,1] => 11111 => 11111
[1,1,1,2] => [2,1,1,1] => 10111 => 11100
[1,1,2,1] => [1,2,1,1] => 11011 => 11001
[1,1,3] => [3,1,1] => 10011 => 11010
[1,2,1,1] => [1,1,2,1] => 11101 => 10011
[1,2,2] => [2,2,1] => 10101 => 10000
[1,3,1] => [1,3,1] => 11001 => 10101
[1,4] => [4,1] => 10001 => 10110
[2,1,1,1] => [1,1,1,2] => 11110 => 00111
[2,1,2] => [2,1,2] => 10110 => 00100
[2,2,1] => [1,2,2] => 11010 => 00001
[2,3] => [3,2] => 10010 => 00010
[3,1,1] => [1,1,3] => 11100 => 01011
[3,2] => [2,3] => 10100 => 01000
[4,1] => [1,4] => 11000 => 01101
[5] => [5] => 10000 => 01110
[1,1,1,1,1,1] => [1,1,1,1,1,1] => 111111 => 111111
[1,1,1,1,2] => [2,1,1,1,1] => 101111 => 111100
[1,1,1,2,1] => [1,2,1,1,1] => 110111 => 111001
[1,1,1,3] => [3,1,1,1] => 100111 => 111010
[1,1,2,1,1] => [1,1,2,1,1] => 111011 => 110011
[1,1,2,2] => [2,2,1,1] => 101011 => 110000
[1,1,3,1] => [1,3,1,1] => 110011 => 110101
[1,1,4] => [4,1,1] => 100011 => 110110
[1,2,1,1,1] => [1,1,1,2,1] => 111101 => 100111
[1,2,1,2] => [2,1,2,1] => 101101 => 100100
[1,2,2,1] => [1,2,2,1] => 110101 => 100001
[1,2,3] => [3,2,1] => 100101 => 100010
[1,3,1,1] => [1,1,3,1] => 111001 => 101011
[1,3,2] => [2,3,1] => 101001 => 101000
[1,4,1] => [1,4,1] => 110001 => 101101
[1,5] => [5,1] => 100001 => 101110
[2,1,1,1,1] => [1,1,1,1,2] => 111110 => 001111
[2,1,1,2] => [2,1,1,2] => 101110 => 001100
[2,1,2,1] => [1,2,1,2] => 110110 => 001001
[2,1,3] => [3,1,2] => 100110 => 001010
[2,2,1,1] => [1,1,2,2] => 111010 => 000011
[2,2,2] => [2,2,2] => 101010 => 000000
[2,3,1] => [1,3,2] => 110010 => 000101
[2,4] => [4,2] => 100010 => 000110
[3,1,1,1] => [1,1,1,3] => 111100 => 010111
[3,1,2] => [2,1,3] => 101100 => 010100
[3,2,1] => [1,2,3] => 110100 => 010001
[3,3] => [3,3] => 100100 => 010010
[4,1,1] => [1,1,4] => 111000 => 011011
[4,2] => [2,4] => 101000 => 011000
[5,1] => [1,5] => 110000 => 011101
[6] => [6] => 100000 => 011110
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1111111 => 1111111
[1,1,1,1,1,2] => [2,1,1,1,1,1] => 1011111 => 1111100
[1,1,1,1,2,1] => [1,2,1,1,1,1] => 1101111 => 1111001
[1,1,1,1,3] => [3,1,1,1,1] => 1001111 => 1111010
[1,1,1,2,1,1] => [1,1,2,1,1,1] => 1110111 => 1110011
[1,1,1,2,2] => [2,2,1,1,1] => 1010111 => 1110000
[1,1,1,3,1] => [1,3,1,1,1] => 1100111 => 1110101
[1,1,1,4] => [4,1,1,1] => 1000111 => 1110110
[1,1,2,1,1,1] => [1,1,1,2,1,1] => 1111011 => 1100111
[1,1,2,1,2] => [2,1,2,1,1] => 1011011 => 1100100
[1,1,2,2,1] => [1,2,2,1,1] => 1101011 => 1100001
[1,1,2,3] => [3,2,1,1] => 1001011 => 1100010
[1,1,3,1,1] => [1,1,3,1,1] => 1110011 => 1101011
[1,1,3,2] => [2,3,1,1] => 1010011 => 1101000
[1,1,4,1] => [1,4,1,1] => 1100011 => 1101101
[1,1,5] => [5,1,1] => 1000011 => 1101110
[1,2,1,1,1,1] => [1,1,1,1,2,1] => 1111101 => 1001111
[1,2,1,1,2] => [2,1,1,2,1] => 1011101 => 1001100
[1,2,1,2,1] => [1,2,1,2,1] => 1101101 => 1001001
[1,2,1,3] => [3,1,2,1] => 1001101 => 1001010
[1,2,2,1,1] => [1,1,2,2,1] => 1110101 => 1000011
[1,2,2,2] => [2,2,2,1] => 1010101 => 1000000
[1,2,3,1] => [1,3,2,1] => 1100101 => 1000101
[1,2,4] => [4,2,1] => 1000101 => 1000110
[1,3,1,1,1] => [1,1,1,3,1] => 1111001 => 1010111
[1,3,1,2] => [2,1,3,1] => 1011001 => 1010100
[1,3,2,1] => [1,2,3,1] => 1101001 => 1010001
[1,3,3] => [3,3,1] => 1001001 => 1010010
[1,4,1,1] => [1,1,4,1] => 1110001 => 1011011
[1,4,2] => [2,4,1] => 1010001 => 1011000
[1,5,1] => [1,5,1] => 1100001 => 1011101
[1,6] => [6,1] => 1000001 => 1011110
[2,1,1,1,1,1] => [1,1,1,1,1,2] => 1111110 => 0011111
[2,1,1,1,2] => [2,1,1,1,2] => 1011110 => 0011100
[2,1,1,2,1] => [1,2,1,1,2] => 1101110 => 0011001
[2,1,1,3] => [3,1,1,2] => 1001110 => 0011010
[2,1,2,1,1] => [1,1,2,1,2] => 1110110 => 0010011
[2,1,2,2] => [2,2,1,2] => 1010110 => 0010000
>>> Load all 739 entries. <<<
[2,1,3,1] => [1,3,1,2] => 1100110 => 0010101
[2,1,4] => [4,1,2] => 1000110 => 0010110
[2,2,1,1,1] => [1,1,1,2,2] => 1111010 => 0000111
[2,2,1,2] => [2,1,2,2] => 1011010 => 0000100
[2,2,2,1] => [1,2,2,2] => 1101010 => 0000001
[2,2,3] => [3,2,2] => 1001010 => 0000010
[2,3,1,1] => [1,1,3,2] => 1110010 => 0001011
[2,3,2] => [2,3,2] => 1010010 => 0001000
[2,4,1] => [1,4,2] => 1100010 => 0001101
[2,5] => [5,2] => 1000010 => 0001110
[3,1,1,1,1] => [1,1,1,1,3] => 1111100 => 0101111
[3,1,1,2] => [2,1,1,3] => 1011100 => 0101100
[3,1,2,1] => [1,2,1,3] => 1101100 => 0101001
[3,1,3] => [3,1,3] => 1001100 => 0101010
[3,2,1,1] => [1,1,2,3] => 1110100 => 0100011
[3,2,2] => [2,2,3] => 1010100 => 0100000
[3,3,1] => [1,3,3] => 1100100 => 0100101
[3,4] => [4,3] => 1000100 => 0100110
[4,1,1,1] => [1,1,1,4] => 1111000 => 0110111
[4,1,2] => [2,1,4] => 1011000 => 0110100
[4,2,1] => [1,2,4] => 1101000 => 0110001
[4,3] => [3,4] => 1001000 => 0110010
[5,1,1] => [1,1,5] => 1110000 => 0111011
[5,2] => [2,5] => 1010000 => 0111000
[6,1] => [1,6] => 1100000 => 0111101
[7] => [7] => 1000000 => 0111110
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 11111111 => 11111111
[1,1,1,1,1,1,2] => [2,1,1,1,1,1,1] => 10111111 => 11111100
[1,1,1,1,1,2,1] => [1,2,1,1,1,1,1] => 11011111 => 11111001
[1,1,1,1,1,3] => [3,1,1,1,1,1] => 10011111 => 11111010
[1,1,1,1,2,1,1] => [1,1,2,1,1,1,1] => 11101111 => 11110011
[1,1,1,1,2,2] => [2,2,1,1,1,1] => 10101111 => 11110000
[1,1,1,1,3,1] => [1,3,1,1,1,1] => 11001111 => 11110101
[1,1,1,1,4] => [4,1,1,1,1] => 10001111 => 11110110
[1,1,1,2,1,1,1] => [1,1,1,2,1,1,1] => 11110111 => 11100111
[1,1,1,2,1,2] => [2,1,2,1,1,1] => 10110111 => 11100100
[1,1,1,2,2,1] => [1,2,2,1,1,1] => 11010111 => 11100001
[1,1,1,2,3] => [3,2,1,1,1] => 10010111 => 11100010
[1,1,1,3,1,1] => [1,1,3,1,1,1] => 11100111 => 11101011
[1,1,1,3,2] => [2,3,1,1,1] => 10100111 => 11101000
[1,1,1,4,1] => [1,4,1,1,1] => 11000111 => 11101101
[1,1,1,5] => [5,1,1,1] => 10000111 => 11101110
[1,1,2,1,1,1,1] => [1,1,1,1,2,1,1] => 11111011 => 11001111
[1,1,2,1,1,2] => [2,1,1,2,1,1] => 10111011 => 11001100
[1,1,2,1,2,1] => [1,2,1,2,1,1] => 11011011 => 11001001
[1,1,2,1,3] => [3,1,2,1,1] => 10011011 => 11001010
[1,1,2,2,1,1] => [1,1,2,2,1,1] => 11101011 => 11000011
[1,1,2,2,2] => [2,2,2,1,1] => 10101011 => 11000000
[1,1,2,3,1] => [1,3,2,1,1] => 11001011 => 11000101
[1,1,2,4] => [4,2,1,1] => 10001011 => 11000110
[1,1,3,1,1,1] => [1,1,1,3,1,1] => 11110011 => 11010111
[1,1,3,1,2] => [2,1,3,1,1] => 10110011 => 11010100
[1,1,3,2,1] => [1,2,3,1,1] => 11010011 => 11010001
[1,1,3,3] => [3,3,1,1] => 10010011 => 11010010
[1,1,4,1,1] => [1,1,4,1,1] => 11100011 => 11011011
[1,1,4,2] => [2,4,1,1] => 10100011 => 11011000
[1,1,5,1] => [1,5,1,1] => 11000011 => 11011101
[1,1,6] => [6,1,1] => 10000011 => 11011110
[1,2,1,1,1,1,1] => [1,1,1,1,1,2,1] => 11111101 => 10011111
[1,2,1,1,1,2] => [2,1,1,1,2,1] => 10111101 => 10011100
[1,2,1,1,2,1] => [1,2,1,1,2,1] => 11011101 => 10011001
[1,2,1,1,3] => [3,1,1,2,1] => 10011101 => 10011010
[1,2,1,2,1,1] => [1,1,2,1,2,1] => 11101101 => 10010011
[1,2,1,2,2] => [2,2,1,2,1] => 10101101 => 10010000
[1,2,1,3,1] => [1,3,1,2,1] => 11001101 => 10010101
[1,2,1,4] => [4,1,2,1] => 10001101 => 10010110
[1,2,2,1,1,1] => [1,1,1,2,2,1] => 11110101 => 10000111
[1,2,2,1,2] => [2,1,2,2,1] => 10110101 => 10000100
[1,2,2,2,1] => [1,2,2,2,1] => 11010101 => 10000001
[1,2,2,3] => [3,2,2,1] => 10010101 => 10000010
[1,2,3,1,1] => [1,1,3,2,1] => 11100101 => 10001011
[1,2,3,2] => [2,3,2,1] => 10100101 => 10001000
[1,2,4,1] => [1,4,2,1] => 11000101 => 10001101
[1,2,5] => [5,2,1] => 10000101 => 10001110
[1,3,1,1,1,1] => [1,1,1,1,3,1] => 11111001 => 10101111
[1,3,1,1,2] => [2,1,1,3,1] => 10111001 => 10101100
[1,3,1,2,1] => [1,2,1,3,1] => 11011001 => 10101001
[1,3,1,3] => [3,1,3,1] => 10011001 => 10101010
[1,3,2,1,1] => [1,1,2,3,1] => 11101001 => 10100011
[1,3,2,2] => [2,2,3,1] => 10101001 => 10100000
[1,3,3,1] => [1,3,3,1] => 11001001 => 10100101
[1,3,4] => [4,3,1] => 10001001 => 10100110
[1,4,1,1,1] => [1,1,1,4,1] => 11110001 => 10110111
[1,4,1,2] => [2,1,4,1] => 10110001 => 10110100
[1,4,2,1] => [1,2,4,1] => 11010001 => 10110001
[1,4,3] => [3,4,1] => 10010001 => 10110010
[1,5,1,1] => [1,1,5,1] => 11100001 => 10111011
[1,5,2] => [2,5,1] => 10100001 => 10111000
[1,6,1] => [1,6,1] => 11000001 => 10111101
[1,7] => [7,1] => 10000001 => 10111110
[2,1,1,1,1,1,1] => [1,1,1,1,1,1,2] => 11111110 => 00111111
[2,1,1,1,1,2] => [2,1,1,1,1,2] => 10111110 => 00111100
[2,1,1,1,2,1] => [1,2,1,1,1,2] => 11011110 => 00111001
[2,1,1,1,3] => [3,1,1,1,2] => 10011110 => 00111010
[2,1,1,2,1,1] => [1,1,2,1,1,2] => 11101110 => 00110011
[2,1,1,2,2] => [2,2,1,1,2] => 10101110 => 00110000
[2,1,1,3,1] => [1,3,1,1,2] => 11001110 => 00110101
[2,1,1,4] => [4,1,1,2] => 10001110 => 00110110
[2,1,2,1,1,1] => [1,1,1,2,1,2] => 11110110 => 00100111
[2,1,2,1,2] => [2,1,2,1,2] => 10110110 => 00100100
[2,1,2,2,1] => [1,2,2,1,2] => 11010110 => 00100001
[2,1,2,3] => [3,2,1,2] => 10010110 => 00100010
[2,1,3,1,1] => [1,1,3,1,2] => 11100110 => 00101011
[2,1,3,2] => [2,3,1,2] => 10100110 => 00101000
[2,1,4,1] => [1,4,1,2] => 11000110 => 00101101
[2,1,5] => [5,1,2] => 10000110 => 00101110
[2,2,1,1,1,1] => [1,1,1,1,2,2] => 11111010 => 00001111
[2,2,1,1,2] => [2,1,1,2,2] => 10111010 => 00001100
[2,2,1,2,1] => [1,2,1,2,2] => 11011010 => 00001001
[2,2,1,3] => [3,1,2,2] => 10011010 => 00001010
[2,2,2,1,1] => [1,1,2,2,2] => 11101010 => 00000011
[2,2,2,2] => [2,2,2,2] => 10101010 => 00000000
[2,2,3,1] => [1,3,2,2] => 11001010 => 00000101
[2,2,4] => [4,2,2] => 10001010 => 00000110
[2,3,1,1,1] => [1,1,1,3,2] => 11110010 => 00010111
[2,3,1,2] => [2,1,3,2] => 10110010 => 00010100
[2,3,2,1] => [1,2,3,2] => 11010010 => 00010001
[2,3,3] => [3,3,2] => 10010010 => 00010010
[2,4,1,1] => [1,1,4,2] => 11100010 => 00011011
[2,4,2] => [2,4,2] => 10100010 => 00011000
[2,5,1] => [1,5,2] => 11000010 => 00011101
[2,6] => [6,2] => 10000010 => 00011110
[3,1,1,1,1,1] => [1,1,1,1,1,3] => 11111100 => 01011111
[3,1,1,1,2] => [2,1,1,1,3] => 10111100 => 01011100
[3,1,1,2,1] => [1,2,1,1,3] => 11011100 => 01011001
[3,1,1,3] => [3,1,1,3] => 10011100 => 01011010
[3,1,2,1,1] => [1,1,2,1,3] => 11101100 => 01010011
[3,1,2,2] => [2,2,1,3] => 10101100 => 01010000
[3,1,3,1] => [1,3,1,3] => 11001100 => 01010101
[3,1,4] => [4,1,3] => 10001100 => 01010110
[3,2,1,1,1] => [1,1,1,2,3] => 11110100 => 01000111
[3,2,1,2] => [2,1,2,3] => 10110100 => 01000100
[3,2,2,1] => [1,2,2,3] => 11010100 => 01000001
[3,2,3] => [3,2,3] => 10010100 => 01000010
[3,3,1,1] => [1,1,3,3] => 11100100 => 01001011
[3,3,2] => [2,3,3] => 10100100 => 01001000
[3,4,1] => [1,4,3] => 11000100 => 01001101
[3,5] => [5,3] => 10000100 => 01001110
[4,1,1,1,1] => [1,1,1,1,4] => 11111000 => 01101111
[4,1,1,2] => [2,1,1,4] => 10111000 => 01101100
[4,1,2,1] => [1,2,1,4] => 11011000 => 01101001
[4,1,3] => [3,1,4] => 10011000 => 01101010
[4,2,1,1] => [1,1,2,4] => 11101000 => 01100011
[4,2,2] => [2,2,4] => 10101000 => 01100000
[4,3,1] => [1,3,4] => 11001000 => 01100101
[4,4] => [4,4] => 10001000 => 01100110
[5,1,1,1] => [1,1,1,5] => 11110000 => 01110111
[5,1,2] => [2,1,5] => 10110000 => 01110100
[5,2,1] => [1,2,5] => 11010000 => 01110001
[5,3] => [3,5] => 10010000 => 01110010
[6,1,1] => [1,1,6] => 11100000 => 01111011
[6,2] => [2,6] => 10100000 => 01111000
[7,1] => [1,7] => 11000000 => 01111101
[8] => [8] => 10000000 => 01111110
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 111111111 => 111111111
[1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1] => 101111111 => 111111100
[1,1,1,1,1,1,2,1] => [1,2,1,1,1,1,1,1] => 110111111 => 111111001
[1,1,1,1,1,1,3] => [3,1,1,1,1,1,1] => 100111111 => 111111010
[1,1,1,1,1,2,1,1] => [1,1,2,1,1,1,1,1] => 111011111 => 111110011
[1,1,1,1,1,2,2] => [2,2,1,1,1,1,1] => 101011111 => 111110000
[1,1,1,1,1,3,1] => [1,3,1,1,1,1,1] => 110011111 => 111110101
[1,1,1,1,1,4] => [4,1,1,1,1,1] => 100011111 => 111110110
[1,1,1,1,2,1,1,1] => [1,1,1,2,1,1,1,1] => 111101111 => 111100111
[1,1,1,1,2,1,2] => [2,1,2,1,1,1,1] => 101101111 => 111100100
[1,1,1,1,2,2,1] => [1,2,2,1,1,1,1] => 110101111 => 111100001
[1,1,1,1,2,3] => [3,2,1,1,1,1] => 100101111 => 111100010
[1,1,1,1,3,1,1] => [1,1,3,1,1,1,1] => 111001111 => 111101011
[1,1,1,1,3,2] => [2,3,1,1,1,1] => 101001111 => 111101000
[1,1,1,1,4,1] => [1,4,1,1,1,1] => 110001111 => 111101101
[1,1,1,1,5] => [5,1,1,1,1] => 100001111 => 111101110
[1,1,1,2,1,1,1,1] => [1,1,1,1,2,1,1,1] => 111110111 => 111001111
[1,1,1,2,1,1,2] => [2,1,1,2,1,1,1] => 101110111 => 111001100
[1,1,1,2,1,2,1] => [1,2,1,2,1,1,1] => 110110111 => 111001001
[1,1,1,2,1,3] => [3,1,2,1,1,1] => 100110111 => 111001010
[1,1,1,2,2,1,1] => [1,1,2,2,1,1,1] => 111010111 => 111000011
[1,1,1,2,2,2] => [2,2,2,1,1,1] => 101010111 => 111000000
[1,1,1,2,3,1] => [1,3,2,1,1,1] => 110010111 => 111000101
[1,1,1,2,4] => [4,2,1,1,1] => 100010111 => 111000110
[1,1,1,3,1,1,1] => [1,1,1,3,1,1,1] => 111100111 => 111010111
[1,1,1,3,1,2] => [2,1,3,1,1,1] => 101100111 => 111010100
[1,1,1,3,2,1] => [1,2,3,1,1,1] => 110100111 => 111010001
[1,1,1,3,3] => [3,3,1,1,1] => 100100111 => 111010010
[1,1,1,4,1,1] => [1,1,4,1,1,1] => 111000111 => 111011011
[1,1,1,4,2] => [2,4,1,1,1] => 101000111 => 111011000
[1,1,1,5,1] => [1,5,1,1,1] => 110000111 => 111011101
[1,1,1,6] => [6,1,1,1] => 100000111 => 111011110
[1,1,2,1,1,1,1,1] => [1,1,1,1,1,2,1,1] => 111111011 => 110011111
[1,1,2,1,1,1,2] => [2,1,1,1,2,1,1] => 101111011 => 110011100
[1,1,2,1,1,2,1] => [1,2,1,1,2,1,1] => 110111011 => 110011001
[1,1,2,1,1,3] => [3,1,1,2,1,1] => 100111011 => 110011010
[1,1,2,1,2,1,1] => [1,1,2,1,2,1,1] => 111011011 => 110010011
[1,1,2,1,2,2] => [2,2,1,2,1,1] => 101011011 => 110010000
[1,1,2,1,3,1] => [1,3,1,2,1,1] => 110011011 => 110010101
[1,1,2,1,4] => [4,1,2,1,1] => 100011011 => 110010110
[1,1,2,2,1,1,1] => [1,1,1,2,2,1,1] => 111101011 => 110000111
[1,1,2,2,1,2] => [2,1,2,2,1,1] => 101101011 => 110000100
[1,1,2,2,2,1] => [1,2,2,2,1,1] => 110101011 => 110000001
[1,1,2,2,3] => [3,2,2,1,1] => 100101011 => 110000010
[1,1,2,3,1,1] => [1,1,3,2,1,1] => 111001011 => 110001011
[1,1,2,3,2] => [2,3,2,1,1] => 101001011 => 110001000
[1,1,2,4,1] => [1,4,2,1,1] => 110001011 => 110001101
[1,1,2,5] => [5,2,1,1] => 100001011 => 110001110
[1,1,3,1,1,1,1] => [1,1,1,1,3,1,1] => 111110011 => 110101111
[1,1,3,1,1,2] => [2,1,1,3,1,1] => 101110011 => 110101100
[1,1,3,1,2,1] => [1,2,1,3,1,1] => 110110011 => 110101001
[1,1,3,1,3] => [3,1,3,1,1] => 100110011 => 110101010
[1,1,3,2,1,1] => [1,1,2,3,1,1] => 111010011 => 110100011
[1,1,3,2,2] => [2,2,3,1,1] => 101010011 => 110100000
[1,1,3,3,1] => [1,3,3,1,1] => 110010011 => 110100101
[1,1,3,4] => [4,3,1,1] => 100010011 => 110100110
[1,1,4,1,1,1] => [1,1,1,4,1,1] => 111100011 => 110110111
[1,1,4,1,2] => [2,1,4,1,1] => 101100011 => 110110100
[1,1,4,2,1] => [1,2,4,1,1] => 110100011 => 110110001
[1,1,4,3] => [3,4,1,1] => 100100011 => 110110010
[1,1,5,1,1] => [1,1,5,1,1] => 111000011 => 110111011
[1,1,5,2] => [2,5,1,1] => 101000011 => 110111000
[1,1,6,1] => [1,6,1,1] => 110000011 => 110111101
[1,1,7] => [7,1,1] => 100000011 => 110111110
[1,2,1,1,1,1,1,1] => [1,1,1,1,1,1,2,1] => 111111101 => 100111111
[1,2,1,1,1,1,2] => [2,1,1,1,1,2,1] => 101111101 => 100111100
[1,2,1,1,1,2,1] => [1,2,1,1,1,2,1] => 110111101 => 100111001
[1,2,1,1,1,3] => [3,1,1,1,2,1] => 100111101 => 100111010
[1,2,1,1,2,1,1] => [1,1,2,1,1,2,1] => 111011101 => 100110011
[1,2,1,1,2,2] => [2,2,1,1,2,1] => 101011101 => 100110000
[1,2,1,1,3,1] => [1,3,1,1,2,1] => 110011101 => 100110101
[1,2,1,1,4] => [4,1,1,2,1] => 100011101 => 100110110
[1,2,1,2,1,1,1] => [1,1,1,2,1,2,1] => 111101101 => 100100111
[1,2,1,2,1,2] => [2,1,2,1,2,1] => 101101101 => 100100100
[1,2,1,2,2,1] => [1,2,2,1,2,1] => 110101101 => 100100001
[1,2,1,2,3] => [3,2,1,2,1] => 100101101 => 100100010
[1,2,1,3,1,1] => [1,1,3,1,2,1] => 111001101 => 100101011
[1,2,1,3,2] => [2,3,1,2,1] => 101001101 => 100101000
[1,2,1,4,1] => [1,4,1,2,1] => 110001101 => 100101101
[1,2,1,5] => [5,1,2,1] => 100001101 => 100101110
[1,2,2,1,1,1,1] => [1,1,1,1,2,2,1] => 111110101 => 100001111
[1,2,2,1,1,2] => [2,1,1,2,2,1] => 101110101 => 100001100
[1,2,2,1,2,1] => [1,2,1,2,2,1] => 110110101 => 100001001
[1,2,2,1,3] => [3,1,2,2,1] => 100110101 => 100001010
[1,2,2,2,1,1] => [1,1,2,2,2,1] => 111010101 => 100000011
[1,2,2,2,2] => [2,2,2,2,1] => 101010101 => 100000000
[1,2,2,3,1] => [1,3,2,2,1] => 110010101 => 100000101
[1,2,2,4] => [4,2,2,1] => 100010101 => 100000110
[1,2,3,1,1,1] => [1,1,1,3,2,1] => 111100101 => 100010111
[1,2,3,1,2] => [2,1,3,2,1] => 101100101 => 100010100
[1,2,3,2,1] => [1,2,3,2,1] => 110100101 => 100010001
[1,2,3,3] => [3,3,2,1] => 100100101 => 100010010
[1,2,4,1,1] => [1,1,4,2,1] => 111000101 => 100011011
[1,2,4,2] => [2,4,2,1] => 101000101 => 100011000
[1,2,5,1] => [1,5,2,1] => 110000101 => 100011101
[1,2,6] => [6,2,1] => 100000101 => 100011110
[1,3,1,1,1,1,1] => [1,1,1,1,1,3,1] => 111111001 => 101011111
[1,3,1,1,1,2] => [2,1,1,1,3,1] => 101111001 => 101011100
[1,3,1,1,2,1] => [1,2,1,1,3,1] => 110111001 => 101011001
[1,3,1,1,3] => [3,1,1,3,1] => 100111001 => 101011010
[1,3,1,2,1,1] => [1,1,2,1,3,1] => 111011001 => 101010011
[1,3,1,2,2] => [2,2,1,3,1] => 101011001 => 101010000
[1,3,1,3,1] => [1,3,1,3,1] => 110011001 => 101010101
[1,3,1,4] => [4,1,3,1] => 100011001 => 101010110
[1,3,2,1,1,1] => [1,1,1,2,3,1] => 111101001 => 101000111
[1,3,2,1,2] => [2,1,2,3,1] => 101101001 => 101000100
[1,3,2,2,1] => [1,2,2,3,1] => 110101001 => 101000001
[1,3,2,3] => [3,2,3,1] => 100101001 => 101000010
[1,3,3,1,1] => [1,1,3,3,1] => 111001001 => 101001011
[1,3,3,2] => [2,3,3,1] => 101001001 => 101001000
[1,3,4,1] => [1,4,3,1] => 110001001 => 101001101
[1,3,5] => [5,3,1] => 100001001 => 101001110
[1,4,1,1,1,1] => [1,1,1,1,4,1] => 111110001 => 101101111
[1,4,1,1,2] => [2,1,1,4,1] => 101110001 => 101101100
[1,4,1,2,1] => [1,2,1,4,1] => 110110001 => 101101001
[1,4,1,3] => [3,1,4,1] => 100110001 => 101101010
[1,4,2,1,1] => [1,1,2,4,1] => 111010001 => 101100011
[1,4,2,2] => [2,2,4,1] => 101010001 => 101100000
[1,4,3,1] => [1,3,4,1] => 110010001 => 101100101
[1,4,4] => [4,4,1] => 100010001 => 101100110
[1,5,1,1,1] => [1,1,1,5,1] => 111100001 => 101110111
[1,5,1,2] => [2,1,5,1] => 101100001 => 101110100
[1,5,2,1] => [1,2,5,1] => 110100001 => 101110001
[1,5,3] => [3,5,1] => 100100001 => 101110010
[1,6,1,1] => [1,1,6,1] => 111000001 => 101111011
[1,6,2] => [2,6,1] => 101000001 => 101111000
[1,7,1] => [1,7,1] => 110000001 => 101111101
[1,8] => [8,1] => 100000001 => 101111110
[2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,2] => 111111110 => 001111111
[2,1,1,1,1,1,2] => [2,1,1,1,1,1,2] => 101111110 => 001111100
[2,1,1,1,1,2,1] => [1,2,1,1,1,1,2] => 110111110 => 001111001
[2,1,1,1,1,3] => [3,1,1,1,1,2] => 100111110 => 001111010
[2,1,1,1,2,1,1] => [1,1,2,1,1,1,2] => 111011110 => 001110011
[2,1,1,1,2,2] => [2,2,1,1,1,2] => 101011110 => 001110000
[2,1,1,1,3,1] => [1,3,1,1,1,2] => 110011110 => 001110101
[2,1,1,1,4] => [4,1,1,1,2] => 100011110 => 001110110
[2,1,1,2,1,1,1] => [1,1,1,2,1,1,2] => 111101110 => 001100111
[2,1,1,2,1,2] => [2,1,2,1,1,2] => 101101110 => 001100100
[2,1,1,2,2,1] => [1,2,2,1,1,2] => 110101110 => 001100001
[2,1,1,2,3] => [3,2,1,1,2] => 100101110 => 001100010
[2,1,1,3,1,1] => [1,1,3,1,1,2] => 111001110 => 001101011
[2,1,1,3,2] => [2,3,1,1,2] => 101001110 => 001101000
[2,1,1,4,1] => [1,4,1,1,2] => 110001110 => 001101101
[2,1,1,5] => [5,1,1,2] => 100001110 => 001101110
[2,1,2,1,1,1,1] => [1,1,1,1,2,1,2] => 111110110 => 001001111
[2,1,2,1,1,2] => [2,1,1,2,1,2] => 101110110 => 001001100
[2,1,2,1,2,1] => [1,2,1,2,1,2] => 110110110 => 001001001
[2,1,2,1,3] => [3,1,2,1,2] => 100110110 => 001001010
[2,1,2,2,1,1] => [1,1,2,2,1,2] => 111010110 => 001000011
[2,1,2,2,2] => [2,2,2,1,2] => 101010110 => 001000000
[2,1,2,3,1] => [1,3,2,1,2] => 110010110 => 001000101
[2,1,2,4] => [4,2,1,2] => 100010110 => 001000110
[2,1,3,1,1,1] => [1,1,1,3,1,2] => 111100110 => 001010111
[2,1,3,1,2] => [2,1,3,1,2] => 101100110 => 001010100
[2,1,3,2,1] => [1,2,3,1,2] => 110100110 => 001010001
[2,1,3,3] => [3,3,1,2] => 100100110 => 001010010
[2,1,4,1,1] => [1,1,4,1,2] => 111000110 => 001011011
[2,1,4,2] => [2,4,1,2] => 101000110 => 001011000
[2,1,5,1] => [1,5,1,2] => 110000110 => 001011101
[2,1,6] => [6,1,2] => 100000110 => 001011110
[2,2,1,1,1,1,1] => [1,1,1,1,1,2,2] => 111111010 => 000011111
[2,2,1,1,1,2] => [2,1,1,1,2,2] => 101111010 => 000011100
[2,2,1,1,2,1] => [1,2,1,1,2,2] => 110111010 => 000011001
[2,2,1,1,3] => [3,1,1,2,2] => 100111010 => 000011010
[2,2,1,2,1,1] => [1,1,2,1,2,2] => 111011010 => 000010011
[2,2,1,2,2] => [2,2,1,2,2] => 101011010 => 000010000
[2,2,1,3,1] => [1,3,1,2,2] => 110011010 => 000010101
[2,2,1,4] => [4,1,2,2] => 100011010 => 000010110
[2,2,2,1,1,1] => [1,1,1,2,2,2] => 111101010 => 000000111
[2,2,2,1,2] => [2,1,2,2,2] => 101101010 => 000000100
[2,2,2,2,1] => [1,2,2,2,2] => 110101010 => 000000001
[2,2,2,3] => [3,2,2,2] => 100101010 => 000000010
[2,2,3,1,1] => [1,1,3,2,2] => 111001010 => 000001011
[2,2,3,2] => [2,3,2,2] => 101001010 => 000001000
[2,2,4,1] => [1,4,2,2] => 110001010 => 000001101
[2,2,5] => [5,2,2] => 100001010 => 000001110
[2,3,1,1,1,1] => [1,1,1,1,3,2] => 111110010 => 000101111
[2,3,1,1,2] => [2,1,1,3,2] => 101110010 => 000101100
[2,3,1,2,1] => [1,2,1,3,2] => 110110010 => 000101001
[2,3,1,3] => [3,1,3,2] => 100110010 => 000101010
[2,3,2,1,1] => [1,1,2,3,2] => 111010010 => 000100011
[2,3,2,2] => [2,2,3,2] => 101010010 => 000100000
[2,3,3,1] => [1,3,3,2] => 110010010 => 000100101
[2,3,4] => [4,3,2] => 100010010 => 000100110
[2,4,1,1,1] => [1,1,1,4,2] => 111100010 => 000110111
[2,4,1,2] => [2,1,4,2] => 101100010 => 000110100
[2,4,2,1] => [1,2,4,2] => 110100010 => 000110001
[2,4,3] => [3,4,2] => 100100010 => 000110010
[2,5,1,1] => [1,1,5,2] => 111000010 => 000111011
[2,5,2] => [2,5,2] => 101000010 => 000111000
[2,6,1] => [1,6,2] => 110000010 => 000111101
[2,7] => [7,2] => 100000010 => 000111110
[3,1,1,1,1,1,1] => [1,1,1,1,1,1,3] => 111111100 => 010111111
[3,1,1,1,1,2] => [2,1,1,1,1,3] => 101111100 => 010111100
[3,1,1,1,2,1] => [1,2,1,1,1,3] => 110111100 => 010111001
[3,1,1,1,3] => [3,1,1,1,3] => 100111100 => 010111010
[3,1,1,2,1,1] => [1,1,2,1,1,3] => 111011100 => 010110011
[3,1,1,2,2] => [2,2,1,1,3] => 101011100 => 010110000
[3,1,1,3,1] => [1,3,1,1,3] => 110011100 => 010110101
[3,1,1,4] => [4,1,1,3] => 100011100 => 010110110
[3,1,2,1,1,1] => [1,1,1,2,1,3] => 111101100 => 010100111
[3,1,2,1,2] => [2,1,2,1,3] => 101101100 => 010100100
[3,1,2,2,1] => [1,2,2,1,3] => 110101100 => 010100001
[3,1,2,3] => [3,2,1,3] => 100101100 => 010100010
[3,1,3,1,1] => [1,1,3,1,3] => 111001100 => 010101011
[3,1,3,2] => [2,3,1,3] => 101001100 => 010101000
[3,1,4,1] => [1,4,1,3] => 110001100 => 010101101
[3,1,5] => [5,1,3] => 100001100 => 010101110
[3,2,1,1,1,1] => [1,1,1,1,2,3] => 111110100 => 010001111
[3,2,1,1,2] => [2,1,1,2,3] => 101110100 => 010001100
[3,2,1,2,1] => [1,2,1,2,3] => 110110100 => 010001001
[3,2,1,3] => [3,1,2,3] => 100110100 => 010001010
[3,2,2,1,1] => [1,1,2,2,3] => 111010100 => 010000011
[3,2,2,2] => [2,2,2,3] => 101010100 => 010000000
[3,2,3,1] => [1,3,2,3] => 110010100 => 010000101
[3,2,4] => [4,2,3] => 100010100 => 010000110
[3,3,1,1,1] => [1,1,1,3,3] => 111100100 => 010010111
[3,3,1,2] => [2,1,3,3] => 101100100 => 010010100
[3,3,2,1] => [1,2,3,3] => 110100100 => 010010001
[3,3,3] => [3,3,3] => 100100100 => 010010010
[3,4,1,1] => [1,1,4,3] => 111000100 => 010011011
[3,4,2] => [2,4,3] => 101000100 => 010011000
[3,5,1] => [1,5,3] => 110000100 => 010011101
[3,6] => [6,3] => 100000100 => 010011110
[4,1,1,1,1,1] => [1,1,1,1,1,4] => 111111000 => 011011111
[4,1,1,1,2] => [2,1,1,1,4] => 101111000 => 011011100
[4,1,1,2,1] => [1,2,1,1,4] => 110111000 => 011011001
[4,1,1,3] => [3,1,1,4] => 100111000 => 011011010
[4,1,2,1,1] => [1,1,2,1,4] => 111011000 => 011010011
[4,1,2,2] => [2,2,1,4] => 101011000 => 011010000
[4,1,3,1] => [1,3,1,4] => 110011000 => 011010101
[4,1,4] => [4,1,4] => 100011000 => 011010110
[4,2,1,1,1] => [1,1,1,2,4] => 111101000 => 011000111
[4,2,1,2] => [2,1,2,4] => 101101000 => 011000100
[4,2,2,1] => [1,2,2,4] => 110101000 => 011000001
[4,2,3] => [3,2,4] => 100101000 => 011000010
[4,3,1,1] => [1,1,3,4] => 111001000 => 011001011
[4,3,2] => [2,3,4] => 101001000 => 011001000
[4,4,1] => [1,4,4] => 110001000 => 011001101
[4,5] => [5,4] => 100001000 => 011001110
[5,1,1,1,1] => [1,1,1,1,5] => 111110000 => 011101111
[5,1,1,2] => [2,1,1,5] => 101110000 => 011101100
[5,1,2,1] => [1,2,1,5] => 110110000 => 011101001
[5,1,3] => [3,1,5] => 100110000 => 011101010
[5,2,1,1] => [1,1,2,5] => 111010000 => 011100011
[5,2,2] => [2,2,5] => 101010000 => 011100000
[5,3,1] => [1,3,5] => 110010000 => 011100101
[5,4] => [4,5] => 100010000 => 011100110
[6,1,1,1] => [1,1,1,6] => 111100000 => 011110111
[6,1,2] => [2,1,6] => 101100000 => 011110100
[6,2,1] => [1,2,6] => 110100000 => 011110001
[6,3] => [3,6] => 100100000 => 011110010
[7,1,1] => [1,1,7] => 111000000 => 011111011
[7,2] => [2,7] => 101000000 => 011111000
[8,1] => [1,8] => 110000000 => 011111101
[9] => [9] => 100000000 => 011111110
[1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111
[1,1,1,1,1,1,1,3] => [3,1,1,1,1,1,1,1] => 1001111111 => 1111111010
[1,1,1,1,1,2,1,2] => [2,1,2,1,1,1,1,1] => 1011011111 => 1111100100
[1,1,1,1,1,2,2,1] => [1,2,2,1,1,1,1,1] => 1101011111 => 1111100001
[1,1,1,1,1,3,1,1] => [1,1,3,1,1,1,1,1] => 1110011111 => 1111101011
[1,1,1,1,1,5] => [5,1,1,1,1,1] => 1000011111 => 1111101110
[1,1,1,2,1,1,1,2] => [2,1,1,1,2,1,1,1] => 1011110111 => 1110011100
[1,1,1,2,1,2,1,1] => [1,1,2,1,2,1,1,1] => 1110110111 => 1110010011
[1,1,1,2,1,4] => [4,1,2,1,1,1] => 1000110111 => 1110010110
[1,1,1,2,2,2,1] => [1,2,2,2,1,1,1] => 1101010111 => 1110000001
[1,1,1,2,3,2] => [2,3,2,1,1,1] => 1010010111 => 1110001000
[1,1,1,3,1,1,1,1] => [1,1,1,1,3,1,1,1] => 1111100111 => 1110101111
[1,1,1,3,1,3] => [3,1,3,1,1,1] => 1001100111 => 1110101010
[1,1,1,4,1,2] => [2,1,4,1,1,1] => 1011000111 => 1110110100
[1,1,1,5,1,1] => [1,1,5,1,1,1] => 1110000111 => 1110111011
[1,1,1,7] => [7,1,1,1] => 1000000111 => 1110111110
[1,1,2,1,1,1,2,1] => [1,2,1,1,1,2,1,1] => 1101111011 => 1100111001
[1,1,2,1,2,2,1] => [1,2,2,1,2,1,1] => 1101011011 => 1100100001
[1,1,2,2,1,1,1,1] => [1,1,1,1,2,2,1,1] => 1111101011 => 1100001111
[1,1,2,2,1,2,1] => [1,2,1,2,2,1,1] => 1101101011 => 1100001001
[1,1,2,2,2,1,1] => [1,1,2,2,2,1,1] => 1110101011 => 1100000011
[1,1,2,2,2,2] => [2,2,2,2,1,1] => 1010101011 => 1100000000
[1,1,2,2,3,1] => [1,3,2,2,1,1] => 1100101011 => 1100000101
[1,1,2,2,4] => [4,2,2,1,1] => 1000101011 => 1100000110
[1,1,2,3,2,1] => [1,2,3,2,1,1] => 1101001011 => 1100010001
[1,1,2,5,1] => [1,5,2,1,1] => 1100001011 => 1100011101
[1,1,3,2,2,1] => [1,2,2,3,1,1] => 1101010011 => 1101000001
[1,1,5,2,1] => [1,2,5,1,1] => 1101000011 => 1101110001
[1,2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,2,1] => 1111111101 => 1001111111
[1,2,1,1,1,1,1,2] => [2,1,1,1,1,1,2,1] => 1011111101 => 1001111100
[1,2,1,1,1,2,1,1] => [1,1,2,1,1,1,2,1] => 1110111101 => 1001110011
[1,2,1,1,1,4] => [4,1,1,1,2,1] => 1000111101 => 1001110110
[1,2,1,1,3,2] => [2,3,1,1,2,1] => 1010011101 => 1001101000
[1,2,1,2,1,1,1,1] => [1,1,1,1,2,1,2,1] => 1111101101 => 1001001111
[1,2,1,2,1,3] => [3,1,2,1,2,1] => 1001101101 => 1001001010
[1,2,1,3,1,2] => [2,1,3,1,2,1] => 1011001101 => 1001010100
[1,2,1,4,1,1] => [1,1,4,1,2,1] => 1110001101 => 1001011011
[1,2,1,6] => [6,1,2,1] => 1000001101 => 1001011110
[1,2,2,1,2,2] => [2,2,1,2,2,1] => 1010110101 => 1000010000
[1,2,2,2,2,1] => [1,2,2,2,2,1] => 1101010101 => 1000000001
[1,2,2,2,3] => [3,2,2,2,1] => 1001010101 => 1000000010
[1,2,2,3,2] => [2,3,2,2,1] => 1010010101 => 1000001000
[1,2,2,4,1] => [1,4,2,2,1] => 1100010101 => 1000001101
[1,2,3,1,1,2] => [2,1,1,3,2,1] => 1011100101 => 1000101100
[1,2,3,1,3] => [3,1,3,2,1] => 1001100101 => 1000101010
[1,2,3,2,1,1] => [1,1,2,3,2,1] => 1110100101 => 1000100011
[1,2,3,2,2] => [2,2,3,2,1] => 1010100101 => 1000100000
[1,2,3,3,1] => [1,3,3,2,1] => 1100100101 => 1000100101
[1,2,3,4] => [4,3,2,1] => 1000100101 => 1000100110
[1,2,4,1,2] => [2,1,4,2,1] => 1011000101 => 1000110100
[1,2,4,2,1] => [1,2,4,2,1] => 1101000101 => 1000110001
[1,2,5,1,1] => [1,1,5,2,1] => 1110000101 => 1000111011
[1,2,5,2] => [2,5,2,1] => 1010000101 => 1000111000
[1,3,1,1,1,1,1,1] => [1,1,1,1,1,1,3,1] => 1111111001 => 1010111111
[1,3,1,1,1,3] => [3,1,1,1,3,1] => 1001111001 => 1010111010
[1,3,1,2,1,2] => [2,1,2,1,3,1] => 1011011001 => 1010100100
[1,3,1,2,3] => [3,2,1,3,1] => 1001011001 => 1010100010
[1,3,1,3,1,1] => [1,1,3,1,3,1] => 1110011001 => 1010101011
[1,3,1,3,2] => [2,3,1,3,1] => 1010011001 => 1010101000
[1,3,1,4,1] => [1,4,1,3,1] => 1100011001 => 1010101101
[1,3,1,5] => [5,1,3,1] => 1000011001 => 1010101110
[1,3,2,1,3] => [3,1,2,3,1] => 1001101001 => 1010001010
[1,3,2,2,2] => [2,2,2,3,1] => 1010101001 => 1010000000
[1,3,2,3,1] => [1,3,2,3,1] => 1100101001 => 1010000101
[1,3,3,1,2] => [2,1,3,3,1] => 1011001001 => 1010010100
[1,3,3,2,1] => [1,2,3,3,1] => 1101001001 => 1010010001
[1,3,4,1,1] => [1,1,4,3,1] => 1110001001 => 1010011011
[1,4,1,1,1,2] => [2,1,1,1,4,1] => 1011110001 => 1011011100
[1,4,1,1,3] => [3,1,1,4,1] => 1001110001 => 1011011010
[1,4,1,2,1,1] => [1,1,2,1,4,1] => 1110110001 => 1011010011
[1,4,1,2,2] => [2,2,1,4,1] => 1010110001 => 1011010000
[1,4,1,3,1] => [1,3,1,4,1] => 1100110001 => 1011010101
[1,4,1,4] => [4,1,4,1] => 1000110001 => 1011010110
[1,4,2,1,2] => [2,1,2,4,1] => 1011010001 => 1011000100
[1,4,2,2,1] => [1,2,2,4,1] => 1101010001 => 1011000001
[1,4,3,1,1] => [1,1,3,4,1] => 1110010001 => 1011001011
[1,4,3,2] => [2,3,4,1] => 1010010001 => 1011001000
[1,5,1,1,1,1] => [1,1,1,1,5,1] => 1111100001 => 1011101111
[1,5,1,1,2] => [2,1,1,5,1] => 1011100001 => 1011101100
[1,5,1,2,1] => [1,2,1,5,1] => 1101100001 => 1011101001
[1,5,1,3] => [3,1,5,1] => 1001100001 => 1011101010
[1,5,2,1,1] => [1,1,2,5,1] => 1110100001 => 1011100011
[1,6,1,1,1] => [1,1,1,6,1] => 1111000001 => 1011110111
[1,6,1,2] => [2,1,6,1] => 1011000001 => 1011110100
[1,7,1,1] => [1,1,7,1] => 1110000001 => 1011111011
[1,9] => [9,1] => 1000000001 => 1011111110
[2,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,2] => 1111111110 => 0011111111
[2,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,2] => 1011111110 => 0011111100
[2,1,1,1,1,1,2,1] => [1,2,1,1,1,1,1,2] => 1101111110 => 0011111001
[2,1,1,1,1,1,3] => [3,1,1,1,1,1,2] => 1001111110 => 0011111010
[2,1,1,1,1,2,1,1] => [1,1,2,1,1,1,1,2] => 1110111110 => 0011110011
[2,1,1,1,1,2,2] => [2,2,1,1,1,1,2] => 1010111110 => 0011110000
[2,1,1,1,1,4] => [4,1,1,1,1,2] => 1000111110 => 0011110110
[2,1,1,1,2,3] => [3,2,1,1,1,2] => 1001011110 => 0011100010
[2,1,1,1,3,1,1] => [1,1,3,1,1,1,2] => 1110011110 => 0011101011
[2,1,1,1,5] => [5,1,1,1,2] => 1000011110 => 0011101110
[2,1,1,2,4] => [4,2,1,1,2] => 1000101110 => 0011000110
[2,1,1,3,2,1] => [1,2,3,1,1,2] => 1101001110 => 0011010001
[2,1,1,3,3] => [3,3,1,1,2] => 1001001110 => 0011010010
[2,1,1,4,2] => [2,4,1,1,2] => 1010001110 => 0011011000
[2,1,1,6] => [6,1,1,2] => 1000001110 => 0011011110
[2,1,2,1,4] => [4,1,2,1,2] => 1000110110 => 0010010110
[2,1,2,2,3] => [3,2,2,1,2] => 1001010110 => 0010000010
[2,1,2,3,2] => [2,3,2,1,2] => 1010010110 => 0010001000
[2,1,2,5] => [5,2,1,2] => 1000010110 => 0010001110
[2,1,3,1,3] => [3,1,3,1,2] => 1001100110 => 0010101010
[2,1,3,2,2] => [2,2,3,1,2] => 1010100110 => 0010100000
[2,1,3,3,1] => [1,3,3,1,2] => 1100100110 => 0010100101
[2,1,7] => [7,1,2] => 1000000110 => 0010111110
[2,2,1,1,4] => [4,1,1,2,2] => 1000111010 => 0000110110
[2,2,1,2,3] => [3,2,1,2,2] => 1001011010 => 0000100010
[2,2,1,3,2] => [2,3,1,2,2] => 1010011010 => 0000101000
[2,2,2,1,3] => [3,1,2,2,2] => 1001101010 => 0000001010
[2,2,2,2,2] => [2,2,2,2,2] => 1010101010 => 0000000000
[2,2,2,3,1] => [1,3,2,2,2] => 1100101010 => 0000000101
[2,2,3,1,2] => [2,1,3,2,2] => 1011001010 => 0000010100
[2,2,3,2,1] => [1,2,3,2,2] => 1101001010 => 0000010001
[2,2,4,1,1] => [1,1,4,2,2] => 1110001010 => 0000011011
[2,2,6] => [6,2,2] => 1000001010 => 0000011110
[2,3,1,2,2] => [2,2,1,3,2] => 1010110010 => 0001010000
[2,3,1,3,1] => [1,3,1,3,2] => 1100110010 => 0001010101
[2,3,2,1,2] => [2,1,2,3,2] => 1011010010 => 0001000100
[2,3,2,2,1] => [1,2,2,3,2] => 1101010010 => 0001000001
[2,3,3,1,1] => [1,1,3,3,2] => 1110010010 => 0001001011
[2,3,4,1] => [1,4,3,2] => 1100010010 => 0001001101
[2,4,1,1,2] => [2,1,1,4,2] => 1011100010 => 0001101100
[2,4,1,2,1] => [1,2,1,4,2] => 1101100010 => 0001101001
[2,4,2,1,1] => [1,1,2,4,2] => 1110100010 => 0001100011
[2,5,1,1,1] => [1,1,1,5,2] => 1111000010 => 0001110111
[2,8] => [8,2] => 1000000010 => 0001111110
[3,1,1,1,4] => [4,1,1,1,3] => 1000111100 => 0101110110
[3,1,2,1,3] => [3,1,2,1,3] => 1001101100 => 0101001010
[3,1,2,2,2] => [2,2,2,1,3] => 1010101100 => 0101000000
[3,1,2,3,1] => [1,3,2,1,3] => 1100101100 => 0101000101
[3,1,3,1,2] => [2,1,3,1,3] => 1011001100 => 0101010100
[3,1,3,2,1] => [1,2,3,1,3] => 1101001100 => 0101010001
[3,1,4,1,1] => [1,1,4,1,3] => 1110001100 => 0101011011
[3,2,1,1,3] => [3,1,1,2,3] => 1001110100 => 0100011010
[3,2,1,2,2] => [2,2,1,2,3] => 1010110100 => 0100010000
[3,2,1,3,1] => [1,3,1,2,3] => 1100110100 => 0100010101
[3,2,2,1,2] => [2,1,2,2,3] => 1011010100 => 0100000100
[3,2,2,2,1] => [1,2,2,2,3] => 1101010100 => 0100000001
[3,2,3,1,1] => [1,1,3,2,3] => 1110010100 => 0100001011
[3,3,1,1,2] => [2,1,1,3,3] => 1011100100 => 0100101100
[3,3,1,2,1] => [1,2,1,3,3] => 1101100100 => 0100101001
[3,3,2,1,1] => [1,1,2,3,3] => 1110100100 => 0100100011
[3,4,1,1,1] => [1,1,1,4,3] => 1111000100 => 0100110111
[3,7] => [7,3] => 1000000100 => 0100111110
[4,1,1,1,3] => [3,1,1,1,4] => 1001111000 => 0110111010
[4,1,1,2,2] => [2,2,1,1,4] => 1010111000 => 0110110000
[4,1,1,3,1] => [1,3,1,1,4] => 1100111000 => 0110110101
[4,1,2,1,2] => [2,1,2,1,4] => 1011011000 => 0110100100
[4,1,2,2,1] => [1,2,2,1,4] => 1101011000 => 0110100001
[4,1,3,1,1] => [1,1,3,1,4] => 1110011000 => 0110101011
[4,2,1,1,2] => [2,1,1,2,4] => 1011101000 => 0110001100
[4,2,1,2,1] => [1,2,1,2,4] => 1101101000 => 0110001001
[4,2,2,1,1] => [1,1,2,2,4] => 1110101000 => 0110000011
[4,3,1,1,1] => [1,1,1,3,4] => 1111001000 => 0110010111
[4,5,1] => [1,5,4] => 1100001000 => 0110011101
[5,1,1,1,2] => [2,1,1,1,5] => 1011110000 => 0111011100
[5,1,1,2,1] => [1,2,1,1,5] => 1101110000 => 0111011001
[5,1,2,1,1] => [1,1,2,1,5] => 1110110000 => 0111010011
[5,2,1,1,1] => [1,1,1,2,5] => 1111010000 => 0111000111
[6,1,1,1,1] => [1,1,1,1,6] => 1111100000 => 0111101111
[10] => [10] => 1000000000 => 0111111110
[1,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1,1] => 111111111111 => 111111111111
[2,2,2,2,2,2] => [2,2,2,2,2,2] => 101010101010 => 000000000000
[2,2,3,3,1,1] => [1,1,3,3,2,2] => 111001001010 => 000001001011
[2,2,3,2,1,2] => [2,1,2,3,2,2] => 101101001010 => 000001000100
[3,3,1,1,2,2] => [2,2,1,1,3,3] => 101011100100 => 010010110000
[3,3,2,2,1,1] => [1,1,2,2,3,3] => 111010100100 => 010010000011
[3,3,2,1,1,2] => [2,1,1,2,3,3] => 101110100100 => 010010001100
[3,2,1,2,2,2] => [2,2,2,1,2,3] => 101010110100 => 010001000000
[3,2,1,1,2,3] => [3,2,1,1,2,3] => 100101110100 => 010001100010
[3,2,2,3,1,1] => [1,1,3,2,2,3] => 111001010100 => 010000001011
[3,2,2,2,1,2] => [2,1,2,2,2,3] => 101101010100 => 010000000100
[3,2,2,1,1,3] => [3,1,1,2,2,3] => 100111010100 => 010000011010
[3,1,2,4,1,1] => [1,1,4,2,1,3] => 111000101100 => 010100011011
[3,1,2,3,1,2] => [2,1,3,2,1,3] => 101100101100 => 010100010100
[3,1,2,1,1,4] => [4,1,1,2,1,3] => 100011101100 => 010100110110
[4,4,1,1,1,1] => [1,1,1,1,4,4] => 111110001000 => 011001101111
[4,3,1,2,1,1] => [1,1,2,1,3,4] => 111011001000 => 011001010011
[4,3,1,1,1,2] => [2,1,1,1,3,4] => 101111001000 => 011001011100
[4,2,1,3,1,1] => [1,1,3,1,2,4] => 111001101000 => 011000101011
[4,2,1,2,1,2] => [2,1,2,1,2,4] => 101101101000 => 011000100100
[4,2,1,1,1,3] => [3,1,1,1,2,4] => 100111101000 => 011000111010
[4,1,1,4,1,1] => [1,1,4,1,1,4] => 111000111000 => 011011011011
[4,1,1,3,1,2] => [2,1,3,1,1,4] => 101100111000 => 011011010100
[4,1,1,1,1,4] => [4,1,1,1,1,4] => 100011111000 => 011011110110
[1,2,2,2,3,2] => [2,3,2,2,2,1] => 101001010101 => 100000001000
[1,2,2,3,3,1] => [1,3,3,2,2,1] => 110010010101 => 100000100101
[1,2,2,3,2,2] => [2,2,3,2,2,1] => 101010010101 => 100000100000
[1,2,3,3,2,1] => [1,2,3,3,2,1] => 110100100101 => 100010010001
[1,2,3,2,3,1] => [1,3,2,3,2,1] => 110010100101 => 100010000101
[1,2,3,2,2,2] => [2,2,2,3,2,1] => 101010100101 => 100010000000
[1,2,5,4] => [4,5,2,1] => 100010000101 => 100011100110
[1,3,3,2,2,1] => [1,2,2,3,3,1] => 110101001001 => 101001000001
[1,3,2,3,2,1] => [1,2,3,2,3,1] => 110100101001 => 101000010001
[1,3,2,2,3,1] => [1,3,2,2,3,1] => 110010101001 => 101000000101
[1,3,2,2,2,2] => [2,2,2,2,3,1] => 101010101001 => 101000000000
[1,4,5,2] => [2,5,4,1] => 101000010001 => 101100111000
[1,4,3,4] => [4,3,4,1] => 100010010001 => 101100100110
[2,3,2,2,2,1] => [1,2,2,2,3,2] => 110101010010 => 000100000001
[2,2,3,2,2,1] => [1,2,2,3,2,2] => 110101001010 => 000001000001
[2,2,2,3,2,1] => [1,2,3,2,2,2] => 110100101010 => 000000010001
[2,2,2,2,3,1] => [1,3,2,2,2,2] => 110010101010 => 000000000101
[1,2,1,2,1,2,1,2] => [2,1,2,1,2,1,2,1] => 101101101101 => 100100100100
[1,2,1,1,2,1,2,2] => [2,2,1,2,1,1,2,1] => 101011011101 => 100110010000
[1,2,2,1,1,1,2,2] => [2,2,1,1,1,2,2,1] => 101011110101 => 100001110000
[1,1,1,2,2,1,2,2] => [2,2,1,2,2,1,1,1] => 101011010111 => 111000010000
[3,3,3,2] => [2,3,3,3] => 10100100100 => 01001001000
[3,4,4] => [4,4,3] => 10001000100 => 01001100110
[2,3,3,3] => [3,3,3,2] => 10010010010 => 00010010010
[4,4,3] => [3,4,4] => 10010001000 => 01100110010
[2,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,2] => 11111111110 => 00111111111
[3,2,2,2,2,1] => [1,2,2,2,2,3] => 110101010100 => 010000000001
[2,1,1,1,1,1,1,2,1] => [1,2,1,1,1,1,1,1,2] => 11011111110 => 00111111001
[3,2,1,2,2,1] => [1,2,2,1,2,3] => 11010110100 => 01000100001
[1,2,2,1,2,2,1,1] => [1,1,2,2,1,2,2,1] => 111010110101 => 100001000011
[1,2,1,1,1,2,1,1,1,1] => [1,1,1,1,2,1,1,1,2,1] => 111110111101 => 100111001111
[1,1,1,2,1,1,1,2,1,1] => [1,1,2,1,1,1,2,1,1,1] => 111011110111 => 111001110011
[1,1,1,2,1,2,1,1,1,1] => [1,1,1,1,2,1,2,1,1,1] => 111110110111 => 111001001111
[11] => [11] => 10000000000 => 01111111110
[2,9] => [9,2] => 10000000010 => 00011111110
[1,2,1,2,1,1,1,1,1,1] => [1,1,1,1,1,1,2,1,2,1] => 111111101101 => 100100111111
[1,2,1,1,1,1,1,2,1,1] => [1,1,2,1,1,1,1,1,2,1] => 111011111101 => 100111110011
[1,1,1,1,1,2,1,2,1,1] => [1,1,2,1,2,1,1,1,1,1] => 111011011111 => 111110010011
[1,2,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,2,1] => 11111111101 => 10011111111
Map
reverse
Description
Return the reversal of a composition.
That is, the composition $(i_1, i_2, \ldots, i_k)$ is sent to $(i_k, i_{k-1}, \ldots, i_1)$.
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 $0$.
Map
flag zeros to zeros
Description
Return a binary word of the same length, such that the number of zeros equals the number of occurrences of $10$ in the word obtained from the original word by prepending the reverse of the complement.
For example, the image of the word $w=1\dots 1$ is $1\dots 1$, because $0\dots 01\dots 1$ has no occurrences of $10$. The words $10\dots 10$ and $010\dots 10$ have image $0\dots 0$.