Identifier
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
Images
[1,0] => 10 => 01 => 00 => 01
[1,0,1,0] => 1010 => 0101 => 1100 => 1001
[1,1,0,0] => 1100 => 0011 => 1110 => 1011
[1,0,1,0,1,0] => 101010 => 010101 => 001100 => 011001
[1,0,1,1,0,0] => 101100 => 010011 => 000100 => 010001
[1,1,0,0,1,0] => 110010 => 001101 => 001110 => 011011
[1,1,0,1,0,0] => 110100 => 001011 => 000110 => 010011
[1,1,1,0,0,0] => 111000 => 000111 => 000010 => 010111
[1,0,1,0,1,0,1,0] => 10101010 => 01010101 => 11001100 => 10011001
[1,0,1,0,1,1,0,0] => 10101100 => 01010011 => 11101100 => 10111001
[1,0,1,1,0,0,1,0] => 10110010 => 01001101 => 11000100 => 10010001
[1,0,1,1,0,1,0,0] => 10110100 => 01001011 => 11100100 => 10110001
[1,0,1,1,1,0,0,0] => 10111000 => 01000111 => 11110100 => 10100001
[1,1,0,0,1,0,1,0] => 11001010 => 00110101 => 11001110 => 10011011
[1,1,0,0,1,1,0,0] => 11001100 => 00110011 => 11101110 => 10111011
[1,1,0,1,0,0,1,0] => 11010010 => 00101101 => 11000110 => 10010011
[1,1,0,1,0,1,0,0] => 11010100 => 00101011 => 11100110 => 10110011
[1,1,0,1,1,0,0,0] => 11011000 => 00100111 => 11110110 => 10100011
[1,1,1,0,0,0,1,0] => 11100010 => 00011101 => 11000010 => 10010111
[1,1,1,0,0,1,0,0] => 11100100 => 00011011 => 11100010 => 10110111
[1,1,1,0,1,0,0,0] => 11101000 => 00010111 => 11110010 => 10100111
[1,1,1,1,0,0,0,0] => 11110000 => 00001111 => 11111010 => 10101111
[1,0,1,0,1,0,1,0,1,0] => 1010101010 => 0101010101 => 0011001100 => 0110011001
[1,0,1,0,1,0,1,1,0,0] => 1010101100 => 0101010011 => 0001001100 => 0100011001
[1,0,1,0,1,1,0,0,1,0] => 1010110010 => 0101001101 => 0011101100 => 0110111001
[1,0,1,0,1,1,0,1,0,0] => 1010110100 => 0101001011 => 0001101100 => 0100111001
[1,0,1,0,1,1,1,0,0,0] => 1010111000 => 0101000111 => 0000101100 => 0101111001
[1,0,1,1,0,0,1,0,1,0] => 1011001010 => 0100110101 => 0011000100 => 0110010001
[1,0,1,1,0,0,1,1,0,0] => 1011001100 => 0100110011 => 0001000100 => 0100010001
[1,0,1,1,0,1,0,0,1,0] => 1011010010 => 0100101101 => 0011100100 => 0110110001
[1,0,1,1,0,1,0,1,0,0] => 1011010100 => 0100101011 => 0001100100 => 0100110001
[1,0,1,1,0,1,1,0,0,0] => 1011011000 => 0100100111 => 0000100100 => 0101110001
[1,0,1,1,1,0,0,0,1,0] => 1011100010 => 0100011101 => 0011110100 => 0110100001
[1,0,1,1,1,0,0,1,0,0] => 1011100100 => 0100011011 => 0001110100 => 0100100001
[1,0,1,1,1,0,1,0,0,0] => 1011101000 => 0100010111 => 0000110100 => 0101100001
[1,0,1,1,1,1,0,0,0,0] => 1011110000 => 0100001111 => 0000010100 => 0101000001
[1,1,0,0,1,0,1,0,1,0] => 1100101010 => 0011010101 => 0011001110 => 0110011011
[1,1,0,0,1,0,1,1,0,0] => 1100101100 => 0011010011 => 0001001110 => 0100011011
[1,1,0,0,1,1,0,0,1,0] => 1100110010 => 0011001101 => 0011101110 => 0110111011
[1,1,0,0,1,1,0,1,0,0] => 1100110100 => 0011001011 => 0001101110 => 0100111011
[1,1,0,0,1,1,1,0,0,0] => 1100111000 => 0011000111 => 0000101110 => 0101111011
[1,1,0,1,0,0,1,0,1,0] => 1101001010 => 0010110101 => 0011000110 => 0110010011
[1,1,0,1,0,0,1,1,0,0] => 1101001100 => 0010110011 => 0001000110 => 0100010011
[1,1,0,1,0,1,0,0,1,0] => 1101010010 => 0010101101 => 0011100110 => 0110110011
[1,1,0,1,0,1,0,1,0,0] => 1101010100 => 0010101011 => 0001100110 => 0100110011
[1,1,0,1,0,1,1,0,0,0] => 1101011000 => 0010100111 => 0000100110 => 0101110011
[1,1,0,1,1,0,0,0,1,0] => 1101100010 => 0010011101 => 0011110110 => 0110100011
[1,1,0,1,1,0,0,1,0,0] => 1101100100 => 0010011011 => 0001110110 => 0100100011
[1,1,0,1,1,0,1,0,0,0] => 1101101000 => 0010010111 => 0000110110 => 0101100011
[1,1,0,1,1,1,0,0,0,0] => 1101110000 => 0010001111 => 0000010110 => 0101000011
[1,1,1,0,0,0,1,0,1,0] => 1110001010 => 0001110101 => 0011000010 => 0110010111
[1,1,1,0,0,0,1,1,0,0] => 1110001100 => 0001110011 => 0001000010 => 0100010111
[1,1,1,0,0,1,0,0,1,0] => 1110010010 => 0001101101 => 0011100010 => 0110110111
[1,1,1,0,0,1,0,1,0,0] => 1110010100 => 0001101011 => 0001100010 => 0100110111
[1,1,1,0,0,1,1,0,0,0] => 1110011000 => 0001100111 => 0000100010 => 0101110111
[1,1,1,0,1,0,0,0,1,0] => 1110100010 => 0001011101 => 0011110010 => 0110100111
[1,1,1,0,1,0,0,1,0,0] => 1110100100 => 0001011011 => 0001110010 => 0100100111
[1,1,1,0,1,0,1,0,0,0] => 1110101000 => 0001010111 => 0000110010 => 0101100111
[1,1,1,0,1,1,0,0,0,0] => 1110110000 => 0001001111 => 0000010010 => 0101000111
[1,1,1,1,0,0,0,0,1,0] => 1111000010 => 0000111101 => 0011111010 => 0110101111
[1,1,1,1,0,0,0,1,0,0] => 1111000100 => 0000111011 => 0001111010 => 0100101111
[1,1,1,1,0,0,1,0,0,0] => 1111001000 => 0000110111 => 0000111010 => 0101101111
[1,1,1,1,0,1,0,0,0,0] => 1111010000 => 0000101111 => 0000011010 => 0101001111
[1,1,1,1,1,0,0,0,0,0] => 1111100000 => 0000011111 => 0000001010 => 0101011111
[1,0,1,0,1,0,1,0,1,0,1,0] => 101010101010 => 010101010101 => 110011001100 => 100110011001
[1,0,1,0,1,0,1,0,1,1,0,0] => 101010101100 => 010101010011 => 111011001100 => 101110011001
[1,0,1,0,1,0,1,1,0,0,1,0] => 101010110010 => 010101001101 => 110001001100 => 100100011001
[1,0,1,0,1,0,1,1,0,1,0,0] => 101010110100 => 010101001011 => 111001001100 => 101100011001
[1,0,1,0,1,0,1,1,1,0,0,0] => 101010111000 => 010101000111 => 111101001100 => 101000011001
[1,0,1,0,1,1,0,0,1,0,1,0] => 101011001010 => 010100110101 => 110011101100 => 100110111001
[1,0,1,0,1,1,0,0,1,1,0,0] => 101011001100 => 010100110011 => 111011101100 => 101110111001
[1,0,1,0,1,1,0,1,0,0,1,0] => 101011010010 => 010100101101 => 110001101100 => 100100111001
[1,0,1,0,1,1,0,1,0,1,0,0] => 101011010100 => 010100101011 => 111001101100 => 101100111001
[1,0,1,0,1,1,0,1,1,0,0,0] => 101011011000 => 010100100111 => 111101101100 => 101000111001
[1,0,1,0,1,1,1,0,0,0,1,0] => 101011100010 => 010100011101 => 110000101100 => 100101111001
[1,0,1,0,1,1,1,0,0,1,0,0] => 101011100100 => 010100011011 => 111000101100 => 101101111001
[1,0,1,0,1,1,1,0,1,0,0,0] => 101011101000 => 010100010111 => 111100101100 => 101001111001
[1,0,1,0,1,1,1,1,0,0,0,0] => 101011110000 => 010100001111 => 111110101100 => 101011111001
[1,0,1,1,0,0,1,0,1,0,1,0] => 101100101010 => 010011010101 => 110011000100 => 100110010001
[1,0,1,1,0,0,1,0,1,1,0,0] => 101100101100 => 010011010011 => 111011000100 => 101110010001
[1,0,1,1,0,0,1,1,0,0,1,0] => 101100110010 => 010011001101 => 110001000100 => 100100010001
[1,0,1,1,0,0,1,1,0,1,0,0] => 101100110100 => 010011001011 => 111001000100 => 101100010001
[1,0,1,1,0,0,1,1,1,0,0,0] => 101100111000 => 010011000111 => 111101000100 => 101000010001
[1,0,1,1,0,1,0,0,1,0,1,0] => 101101001010 => 010010110101 => 110011100100 => 100110110001
[1,0,1,1,0,1,0,0,1,1,0,0] => 101101001100 => 010010110011 => 111011100100 => 101110110001
[1,0,1,1,0,1,0,1,0,0,1,0] => 101101010010 => 010010101101 => 110001100100 => 100100110001
[1,0,1,1,0,1,0,1,0,1,0,0] => 101101010100 => 010010101011 => 111001100100 => 101100110001
[1,0,1,1,0,1,0,1,1,0,0,0] => 101101011000 => 010010100111 => 111101100100 => 101000110001
[1,0,1,1,0,1,1,0,0,0,1,0] => 101101100010 => 010010011101 => 110000100100 => 100101110001
[1,0,1,1,0,1,1,0,0,1,0,0] => 101101100100 => 010010011011 => 111000100100 => 101101110001
[1,0,1,1,0,1,1,0,1,0,0,0] => 101101101000 => 010010010111 => 111100100100 => 101001110001
[1,0,1,1,0,1,1,1,0,0,0,0] => 101101110000 => 010010001111 => 111110100100 => 101011110001
[1,0,1,1,1,0,0,0,1,0,1,0] => 101110001010 => 010001110101 => 110011110100 => 100110100001
[1,0,1,1,1,0,0,0,1,1,0,0] => 101110001100 => 010001110011 => 111011110100 => 101110100001
[1,0,1,1,1,0,0,1,0,0,1,0] => 101110010010 => 010001101101 => 110001110100 => 100100100001
[1,0,1,1,1,0,0,1,0,1,0,0] => 101110010100 => 010001101011 => 111001110100 => 101100100001
[1,0,1,1,1,0,0,1,1,0,0,0] => 101110011000 => 010001100111 => 111101110100 => 101000100001
[1,0,1,1,1,0,1,0,0,0,1,0] => 101110100010 => 010001011101 => 110000110100 => 100101100001
[1,0,1,1,1,0,1,0,0,1,0,0] => 101110100100 => 010001011011 => 111000110100 => 101101100001
[1,0,1,1,1,0,1,0,1,0,0,0] => 101110101000 => 010001010111 => 111100110100 => 101001100001
[1,0,1,1,1,0,1,1,0,0,0,0] => 101110110000 => 010001001111 => 111110110100 => 101011100001
>>> Load all 197 entries. <<<
[1,0,1,1,1,1,0,0,0,0,1,0] => 101111000010 => 010000111101 => 110000010100 => 100101000001
[1,0,1,1,1,1,0,0,0,1,0,0] => 101111000100 => 010000111011 => 111000010100 => 101101000001
[1,0,1,1,1,1,0,0,1,0,0,0] => 101111001000 => 010000110111 => 111100010100 => 101001000001
[1,0,1,1,1,1,0,1,0,0,0,0] => 101111010000 => 010000101111 => 111110010100 => 101011000001
[1,0,1,1,1,1,1,0,0,0,0,0] => 101111100000 => 010000011111 => 111111010100 => 101010000001
[1,1,0,0,1,0,1,0,1,0,1,0] => 110010101010 => 001101010101 => 110011001110 => 100110011011
[1,1,0,0,1,0,1,0,1,1,0,0] => 110010101100 => 001101010011 => 111011001110 => 101110011011
[1,1,0,0,1,0,1,1,0,0,1,0] => 110010110010 => 001101001101 => 110001001110 => 100100011011
[1,1,0,0,1,0,1,1,0,1,0,0] => 110010110100 => 001101001011 => 111001001110 => 101100011011
[1,1,0,0,1,0,1,1,1,0,0,0] => 110010111000 => 001101000111 => 111101001110 => 101000011011
[1,1,0,0,1,1,0,0,1,0,1,0] => 110011001010 => 001100110101 => 110011101110 => 100110111011
[1,1,0,0,1,1,0,0,1,1,0,0] => 110011001100 => 001100110011 => 111011101110 => 101110111011
[1,1,0,0,1,1,0,1,0,0,1,0] => 110011010010 => 001100101101 => 110001101110 => 100100111011
[1,1,0,0,1,1,0,1,0,1,0,0] => 110011010100 => 001100101011 => 111001101110 => 101100111011
[1,1,0,0,1,1,0,1,1,0,0,0] => 110011011000 => 001100100111 => 111101101110 => 101000111011
[1,1,0,0,1,1,1,0,0,0,1,0] => 110011100010 => 001100011101 => 110000101110 => 100101111011
[1,1,0,0,1,1,1,0,0,1,0,0] => 110011100100 => 001100011011 => 111000101110 => 101101111011
[1,1,0,0,1,1,1,0,1,0,0,0] => 110011101000 => 001100010111 => 111100101110 => 101001111011
[1,1,0,0,1,1,1,1,0,0,0,0] => 110011110000 => 001100001111 => 111110101110 => 101011111011
[1,1,0,1,0,0,1,0,1,0,1,0] => 110100101010 => 001011010101 => 110011000110 => 100110010011
[1,1,0,1,0,0,1,0,1,1,0,0] => 110100101100 => 001011010011 => 111011000110 => 101110010011
[1,1,0,1,0,0,1,1,0,0,1,0] => 110100110010 => 001011001101 => 110001000110 => 100100010011
[1,1,0,1,0,0,1,1,0,1,0,0] => 110100110100 => 001011001011 => 111001000110 => 101100010011
[1,1,0,1,0,0,1,1,1,0,0,0] => 110100111000 => 001011000111 => 111101000110 => 101000010011
[1,1,0,1,0,1,0,0,1,0,1,0] => 110101001010 => 001010110101 => 110011100110 => 100110110011
[1,1,0,1,0,1,0,0,1,1,0,0] => 110101001100 => 001010110011 => 111011100110 => 101110110011
[1,1,0,1,0,1,0,1,0,0,1,0] => 110101010010 => 001010101101 => 110001100110 => 100100110011
[1,1,0,1,0,1,0,1,0,1,0,0] => 110101010100 => 001010101011 => 111001100110 => 101100110011
[1,1,0,1,0,1,0,1,1,0,0,0] => 110101011000 => 001010100111 => 111101100110 => 101000110011
[1,1,0,1,0,1,1,0,0,0,1,0] => 110101100010 => 001010011101 => 110000100110 => 100101110011
[1,1,0,1,0,1,1,0,0,1,0,0] => 110101100100 => 001010011011 => 111000100110 => 101101110011
[1,1,0,1,0,1,1,0,1,0,0,0] => 110101101000 => 001010010111 => 111100100110 => 101001110011
[1,1,0,1,0,1,1,1,0,0,0,0] => 110101110000 => 001010001111 => 111110100110 => 101011110011
[1,1,0,1,1,0,0,0,1,0,1,0] => 110110001010 => 001001110101 => 110011110110 => 100110100011
[1,1,0,1,1,0,0,0,1,1,0,0] => 110110001100 => 001001110011 => 111011110110 => 101110100011
[1,1,0,1,1,0,0,1,0,0,1,0] => 110110010010 => 001001101101 => 110001110110 => 100100100011
[1,1,0,1,1,0,0,1,0,1,0,0] => 110110010100 => 001001101011 => 111001110110 => 101100100011
[1,1,0,1,1,0,0,1,1,0,0,0] => 110110011000 => 001001100111 => 111101110110 => 101000100011
[1,1,0,1,1,0,1,0,0,0,1,0] => 110110100010 => 001001011101 => 110000110110 => 100101100011
[1,1,0,1,1,0,1,0,0,1,0,0] => 110110100100 => 001001011011 => 111000110110 => 101101100011
[1,1,0,1,1,0,1,0,1,0,0,0] => 110110101000 => 001001010111 => 111100110110 => 101001100011
[1,1,0,1,1,0,1,1,0,0,0,0] => 110110110000 => 001001001111 => 111110110110 => 101011100011
[1,1,0,1,1,1,0,0,0,0,1,0] => 110111000010 => 001000111101 => 110000010110 => 100101000011
[1,1,0,1,1,1,0,0,0,1,0,0] => 110111000100 => 001000111011 => 111000010110 => 101101000011
[1,1,0,1,1,1,0,0,1,0,0,0] => 110111001000 => 001000110111 => 111100010110 => 101001000011
[1,1,0,1,1,1,0,1,0,0,0,0] => 110111010000 => 001000101111 => 111110010110 => 101011000011
[1,1,0,1,1,1,1,0,0,0,0,0] => 110111100000 => 001000011111 => 111111010110 => 101010000011
[1,1,1,0,0,0,1,0,1,0,1,0] => 111000101010 => 000111010101 => 110011000010 => 100110010111
[1,1,1,0,0,0,1,0,1,1,0,0] => 111000101100 => 000111010011 => 111011000010 => 101110010111
[1,1,1,0,0,0,1,1,0,0,1,0] => 111000110010 => 000111001101 => 110001000010 => 100100010111
[1,1,1,0,0,0,1,1,0,1,0,0] => 111000110100 => 000111001011 => 111001000010 => 101100010111
[1,1,1,0,0,0,1,1,1,0,0,0] => 111000111000 => 000111000111 => 111101000010 => 101000010111
[1,1,1,0,0,1,0,0,1,0,1,0] => 111001001010 => 000110110101 => 110011100010 => 100110110111
[1,1,1,0,0,1,0,0,1,1,0,0] => 111001001100 => 000110110011 => 111011100010 => 101110110111
[1,1,1,0,0,1,0,1,0,0,1,0] => 111001010010 => 000110101101 => 110001100010 => 100100110111
[1,1,1,0,0,1,0,1,0,1,0,0] => 111001010100 => 000110101011 => 111001100010 => 101100110111
[1,1,1,0,0,1,0,1,1,0,0,0] => 111001011000 => 000110100111 => 111101100010 => 101000110111
[1,1,1,0,0,1,1,0,0,0,1,0] => 111001100010 => 000110011101 => 110000100010 => 100101110111
[1,1,1,0,0,1,1,0,0,1,0,0] => 111001100100 => 000110011011 => 111000100010 => 101101110111
[1,1,1,0,0,1,1,0,1,0,0,0] => 111001101000 => 000110010111 => 111100100010 => 101001110111
[1,1,1,0,0,1,1,1,0,0,0,0] => 111001110000 => 000110001111 => 111110100010 => 101011110111
[1,1,1,0,1,0,0,0,1,0,1,0] => 111010001010 => 000101110101 => 110011110010 => 100110100111
[1,1,1,0,1,0,0,0,1,1,0,0] => 111010001100 => 000101110011 => 111011110010 => 101110100111
[1,1,1,0,1,0,0,1,0,0,1,0] => 111010010010 => 000101101101 => 110001110010 => 100100100111
[1,1,1,0,1,0,0,1,0,1,0,0] => 111010010100 => 000101101011 => 111001110010 => 101100100111
[1,1,1,0,1,0,0,1,1,0,0,0] => 111010011000 => 000101100111 => 111101110010 => 101000100111
[1,1,1,0,1,0,1,0,0,0,1,0] => 111010100010 => 000101011101 => 110000110010 => 100101100111
[1,1,1,0,1,0,1,0,0,1,0,0] => 111010100100 => 000101011011 => 111000110010 => 101101100111
[1,1,1,0,1,0,1,0,1,0,0,0] => 111010101000 => 000101010111 => 111100110010 => 101001100111
[1,1,1,0,1,0,1,1,0,0,0,0] => 111010110000 => 000101001111 => 111110110010 => 101011100111
[1,1,1,0,1,1,0,0,0,0,1,0] => 111011000010 => 000100111101 => 110000010010 => 100101000111
[1,1,1,0,1,1,0,0,0,1,0,0] => 111011000100 => 000100111011 => 111000010010 => 101101000111
[1,1,1,0,1,1,0,0,1,0,0,0] => 111011001000 => 000100110111 => 111100010010 => 101001000111
[1,1,1,0,1,1,0,1,0,0,0,0] => 111011010000 => 000100101111 => 111110010010 => 101011000111
[1,1,1,0,1,1,1,0,0,0,0,0] => 111011100000 => 000100011111 => 111111010010 => 101010000111
[1,1,1,1,0,0,0,0,1,0,1,0] => 111100001010 => 000011110101 => 110011111010 => 100110101111
[1,1,1,1,0,0,0,0,1,1,0,0] => 111100001100 => 000011110011 => 111011111010 => 101110101111
[1,1,1,1,0,0,0,1,0,0,1,0] => 111100010010 => 000011101101 => 110001111010 => 100100101111
[1,1,1,1,0,0,0,1,0,1,0,0] => 111100010100 => 000011101011 => 111001111010 => 101100101111
[1,1,1,1,0,0,0,1,1,0,0,0] => 111100011000 => 000011100111 => 111101111010 => 101000101111
[1,1,1,1,0,0,1,0,0,0,1,0] => 111100100010 => 000011011101 => 110000111010 => 100101101111
[1,1,1,1,0,0,1,0,0,1,0,0] => 111100100100 => 000011011011 => 111000111010 => 101101101111
[1,1,1,1,0,0,1,0,1,0,0,0] => 111100101000 => 000011010111 => 111100111010 => 101001101111
[1,1,1,1,0,0,1,1,0,0,0,0] => 111100110000 => 000011001111 => 111110111010 => 101011101111
[1,1,1,1,0,1,0,0,0,0,1,0] => 111101000010 => 000010111101 => 110000011010 => 100101001111
[1,1,1,1,0,1,0,0,0,1,0,0] => 111101000100 => 000010111011 => 111000011010 => 101101001111
[1,1,1,1,0,1,0,0,1,0,0,0] => 111101001000 => 000010110111 => 111100011010 => 101001001111
[1,1,1,1,0,1,0,1,0,0,0,0] => 111101010000 => 000010101111 => 111110011010 => 101011001111
[1,1,1,1,0,1,1,0,0,0,0,0] => 111101100000 => 000010011111 => 111111011010 => 101010001111
[1,1,1,1,1,0,0,0,0,0,1,0] => 111110000010 => 000001111101 => 110000001010 => 100101011111
[1,1,1,1,1,0,0,0,0,1,0,0] => 111110000100 => 000001111011 => 111000001010 => 101101011111
[1,1,1,1,1,0,0,0,1,0,0,0] => 111110001000 => 000001110111 => 111100001010 => 101001011111
[1,1,1,1,1,0,0,1,0,0,0,0] => 111110010000 => 000001101111 => 111110001010 => 101011011111
[1,1,1,1,1,0,1,0,0,0,0,0] => 111110100000 => 000001011111 => 111111001010 => 101010011111
[1,1,1,1,1,1,0,0,0,0,0,0] => 111111000000 => 000000111111 => 111111101010 => 101010111111
[] => => => =>
Map
to binary word
Description
Return the Dyck word as binary word.
Map
complement
Description
Send a binary word to the word obtained by interchanging the two letters.
Map
zeros to flag zeros
Description
Return a binary word of the same length, such that the number of occurrences of $10$ in the word obtained by prepending the reverse of the complement equals the number of $0$s in the original word.
For example, the image of the word $w=1\dots1$ is $1\dots1$, because $w$ has no zeros, and $1\dots1$ is the only word such that prepending the reverse of its complement has no occurrence of the factor $10$.
On the other hand, $0\dots0$ must be mapped to $10\dots10$ if the length is even, and $010\dots10$ if it is odd.
Map
alternating inverse
Description
Sends a binary word $w_1\cdots w_m$ to the binary word $v_1 \cdots v_m$ with $v_i = w_i$ if $i$ is odd and $v_i = 1 - w_i$ if $i$ is even.
This map is used in [1], see Definitions 3.2 and 5.1.