Identifier
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
Images
[1,0] => 10 => 01 => 00
[1,0,1,0] => 1010 => 1001 => 1100
[1,1,0,0] => 1100 => 1011 => 1110
[1,0,1,0,1,0] => 101010 => 011001 => 001100
[1,0,1,1,0,0] => 101100 => 010001 => 000100
[1,1,0,0,1,0] => 110010 => 011011 => 001110
[1,1,0,1,0,0] => 110100 => 010011 => 000110
[1,1,1,0,0,0] => 111000 => 010111 => 000010
[1,0,1,0,1,0,1,0] => 10101010 => 10011001 => 11001100
[1,0,1,0,1,1,0,0] => 10101100 => 10111001 => 11101100
[1,0,1,1,0,0,1,0] => 10110010 => 10010001 => 11000100
[1,0,1,1,0,1,0,0] => 10110100 => 10110001 => 11100100
[1,0,1,1,1,0,0,0] => 10111000 => 10100001 => 11110100
[1,1,0,0,1,0,1,0] => 11001010 => 10011011 => 11001110
[1,1,0,0,1,1,0,0] => 11001100 => 10111011 => 11101110
[1,1,0,1,0,0,1,0] => 11010010 => 10010011 => 11000110
[1,1,0,1,0,1,0,0] => 11010100 => 10110011 => 11100110
[1,1,0,1,1,0,0,0] => 11011000 => 10100011 => 11110110
[1,1,1,0,0,0,1,0] => 11100010 => 10010111 => 11000010
[1,1,1,0,0,1,0,0] => 11100100 => 10110111 => 11100010
[1,1,1,0,1,0,0,0] => 11101000 => 10100111 => 11110010
[1,1,1,1,0,0,0,0] => 11110000 => 10101111 => 11111010
[1,0,1,0,1,0,1,0,1,0] => 1010101010 => 0110011001 => 0011001100
[1,0,1,0,1,0,1,1,0,0] => 1010101100 => 0100011001 => 0001001100
[1,0,1,0,1,1,0,0,1,0] => 1010110010 => 0110111001 => 0011101100
[1,0,1,0,1,1,0,1,0,0] => 1010110100 => 0100111001 => 0001101100
[1,0,1,0,1,1,1,0,0,0] => 1010111000 => 0101111001 => 0000101100
[1,0,1,1,0,0,1,0,1,0] => 1011001010 => 0110010001 => 0011000100
[1,0,1,1,0,0,1,1,0,0] => 1011001100 => 0100010001 => 0001000100
[1,0,1,1,0,1,0,0,1,0] => 1011010010 => 0110110001 => 0011100100
[1,0,1,1,0,1,0,1,0,0] => 1011010100 => 0100110001 => 0001100100
[1,0,1,1,0,1,1,0,0,0] => 1011011000 => 0101110001 => 0000100100
[1,0,1,1,1,0,0,0,1,0] => 1011100010 => 0110100001 => 0011110100
[1,0,1,1,1,0,0,1,0,0] => 1011100100 => 0100100001 => 0001110100
[1,0,1,1,1,0,1,0,0,0] => 1011101000 => 0101100001 => 0000110100
[1,0,1,1,1,1,0,0,0,0] => 1011110000 => 0101000001 => 0000010100
[1,1,0,0,1,0,1,0,1,0] => 1100101010 => 0110011011 => 0011001110
[1,1,0,0,1,0,1,1,0,0] => 1100101100 => 0100011011 => 0001001110
[1,1,0,0,1,1,0,0,1,0] => 1100110010 => 0110111011 => 0011101110
[1,1,0,0,1,1,0,1,0,0] => 1100110100 => 0100111011 => 0001101110
[1,1,0,0,1,1,1,0,0,0] => 1100111000 => 0101111011 => 0000101110
[1,1,0,1,0,0,1,0,1,0] => 1101001010 => 0110010011 => 0011000110
[1,1,0,1,0,0,1,1,0,0] => 1101001100 => 0100010011 => 0001000110
[1,1,0,1,0,1,0,0,1,0] => 1101010010 => 0110110011 => 0011100110
[1,1,0,1,0,1,0,1,0,0] => 1101010100 => 0100110011 => 0001100110
[1,1,0,1,0,1,1,0,0,0] => 1101011000 => 0101110011 => 0000100110
[1,1,0,1,1,0,0,0,1,0] => 1101100010 => 0110100011 => 0011110110
[1,1,0,1,1,0,0,1,0,0] => 1101100100 => 0100100011 => 0001110110
[1,1,0,1,1,0,1,0,0,0] => 1101101000 => 0101100011 => 0000110110
[1,1,0,1,1,1,0,0,0,0] => 1101110000 => 0101000011 => 0000010110
[1,1,1,0,0,0,1,0,1,0] => 1110001010 => 0110010111 => 0011000010
[1,1,1,0,0,0,1,1,0,0] => 1110001100 => 0100010111 => 0001000010
[1,1,1,0,0,1,0,0,1,0] => 1110010010 => 0110110111 => 0011100010
[1,1,1,0,0,1,0,1,0,0] => 1110010100 => 0100110111 => 0001100010
[1,1,1,0,0,1,1,0,0,0] => 1110011000 => 0101110111 => 0000100010
[1,1,1,0,1,0,0,0,1,0] => 1110100010 => 0110100111 => 0011110010
[1,1,1,0,1,0,0,1,0,0] => 1110100100 => 0100100111 => 0001110010
[1,1,1,0,1,0,1,0,0,0] => 1110101000 => 0101100111 => 0000110010
[1,1,1,0,1,1,0,0,0,0] => 1110110000 => 0101000111 => 0000010010
[1,1,1,1,0,0,0,0,1,0] => 1111000010 => 0110101111 => 0011111010
[1,1,1,1,0,0,0,1,0,0] => 1111000100 => 0100101111 => 0001111010
[1,1,1,1,0,0,1,0,0,0] => 1111001000 => 0101101111 => 0000111010
[1,1,1,1,0,1,0,0,0,0] => 1111010000 => 0101001111 => 0000011010
[1,1,1,1,1,0,0,0,0,0] => 1111100000 => 0101011111 => 0000001010
[1,0,1,0,1,0,1,0,1,0,1,0] => 101010101010 => 100110011001 => 110011001100
[1,0,1,0,1,0,1,0,1,1,0,0] => 101010101100 => 101110011001 => 111011001100
[1,0,1,0,1,0,1,1,0,0,1,0] => 101010110010 => 100100011001 => 110001001100
[1,0,1,0,1,0,1,1,0,1,0,0] => 101010110100 => 101100011001 => 111001001100
[1,0,1,0,1,0,1,1,1,0,0,0] => 101010111000 => 101000011001 => 111101001100
[1,0,1,0,1,1,0,0,1,0,1,0] => 101011001010 => 100110111001 => 110011101100
[1,0,1,0,1,1,0,0,1,1,0,0] => 101011001100 => 101110111001 => 111011101100
[1,0,1,0,1,1,0,1,0,0,1,0] => 101011010010 => 100100111001 => 110001101100
[1,0,1,0,1,1,0,1,0,1,0,0] => 101011010100 => 101100111001 => 111001101100
[1,0,1,0,1,1,0,1,1,0,0,0] => 101011011000 => 101000111001 => 111101101100
[1,0,1,0,1,1,1,0,0,0,1,0] => 101011100010 => 100101111001 => 110000101100
[1,0,1,0,1,1,1,0,0,1,0,0] => 101011100100 => 101101111001 => 111000101100
[1,0,1,0,1,1,1,0,1,0,0,0] => 101011101000 => 101001111001 => 111100101100
[1,0,1,0,1,1,1,1,0,0,0,0] => 101011110000 => 101011111001 => 111110101100
[1,0,1,1,0,0,1,0,1,0,1,0] => 101100101010 => 100110010001 => 110011000100
[1,0,1,1,0,0,1,0,1,1,0,0] => 101100101100 => 101110010001 => 111011000100
[1,0,1,1,0,0,1,1,0,0,1,0] => 101100110010 => 100100010001 => 110001000100
[1,0,1,1,0,0,1,1,0,1,0,0] => 101100110100 => 101100010001 => 111001000100
[1,0,1,1,0,0,1,1,1,0,0,0] => 101100111000 => 101000010001 => 111101000100
[1,0,1,1,0,1,0,0,1,0,1,0] => 101101001010 => 100110110001 => 110011100100
[1,0,1,1,0,1,0,0,1,1,0,0] => 101101001100 => 101110110001 => 111011100100
[1,0,1,1,0,1,0,1,0,0,1,0] => 101101010010 => 100100110001 => 110001100100
[1,0,1,1,0,1,0,1,0,1,0,0] => 101101010100 => 101100110001 => 111001100100
[1,0,1,1,0,1,0,1,1,0,0,0] => 101101011000 => 101000110001 => 111101100100
[1,0,1,1,0,1,1,0,0,0,1,0] => 101101100010 => 100101110001 => 110000100100
[1,0,1,1,0,1,1,0,0,1,0,0] => 101101100100 => 101101110001 => 111000100100
[1,0,1,1,0,1,1,0,1,0,0,0] => 101101101000 => 101001110001 => 111100100100
[1,0,1,1,0,1,1,1,0,0,0,0] => 101101110000 => 101011110001 => 111110100100
[1,0,1,1,1,0,0,0,1,0,1,0] => 101110001010 => 100110100001 => 110011110100
[1,0,1,1,1,0,0,0,1,1,0,0] => 101110001100 => 101110100001 => 111011110100
[1,0,1,1,1,0,0,1,0,0,1,0] => 101110010010 => 100100100001 => 110001110100
[1,0,1,1,1,0,0,1,0,1,0,0] => 101110010100 => 101100100001 => 111001110100
[1,0,1,1,1,0,0,1,1,0,0,0] => 101110011000 => 101000100001 => 111101110100
[1,0,1,1,1,0,1,0,0,0,1,0] => 101110100010 => 100101100001 => 110000110100
[1,0,1,1,1,0,1,0,0,1,0,0] => 101110100100 => 101101100001 => 111000110100
[1,0,1,1,1,0,1,0,1,0,0,0] => 101110101000 => 101001100001 => 111100110100
[1,0,1,1,1,0,1,1,0,0,0,0] => 101110110000 => 101011100001 => 111110110100
>>> Load all 205 entries. <<<
[1,0,1,1,1,1,0,0,0,0,1,0] => 101111000010 => 100101000001 => 110000010100
[1,0,1,1,1,1,0,0,0,1,0,0] => 101111000100 => 101101000001 => 111000010100
[1,0,1,1,1,1,0,0,1,0,0,0] => 101111001000 => 101001000001 => 111100010100
[1,0,1,1,1,1,0,1,0,0,0,0] => 101111010000 => 101011000001 => 111110010100
[1,0,1,1,1,1,1,0,0,0,0,0] => 101111100000 => 101010000001 => 111111010100
[1,1,0,0,1,0,1,0,1,0,1,0] => 110010101010 => 100110011011 => 110011001110
[1,1,0,0,1,0,1,0,1,1,0,0] => 110010101100 => 101110011011 => 111011001110
[1,1,0,0,1,0,1,1,0,0,1,0] => 110010110010 => 100100011011 => 110001001110
[1,1,0,0,1,0,1,1,0,1,0,0] => 110010110100 => 101100011011 => 111001001110
[1,1,0,0,1,0,1,1,1,0,0,0] => 110010111000 => 101000011011 => 111101001110
[1,1,0,0,1,1,0,0,1,0,1,0] => 110011001010 => 100110111011 => 110011101110
[1,1,0,0,1,1,0,0,1,1,0,0] => 110011001100 => 101110111011 => 111011101110
[1,1,0,0,1,1,0,1,0,0,1,0] => 110011010010 => 100100111011 => 110001101110
[1,1,0,0,1,1,0,1,0,1,0,0] => 110011010100 => 101100111011 => 111001101110
[1,1,0,0,1,1,0,1,1,0,0,0] => 110011011000 => 101000111011 => 111101101110
[1,1,0,0,1,1,1,0,0,0,1,0] => 110011100010 => 100101111011 => 110000101110
[1,1,0,0,1,1,1,0,0,1,0,0] => 110011100100 => 101101111011 => 111000101110
[1,1,0,0,1,1,1,0,1,0,0,0] => 110011101000 => 101001111011 => 111100101110
[1,1,0,0,1,1,1,1,0,0,0,0] => 110011110000 => 101011111011 => 111110101110
[1,1,0,1,0,0,1,0,1,0,1,0] => 110100101010 => 100110010011 => 110011000110
[1,1,0,1,0,0,1,0,1,1,0,0] => 110100101100 => 101110010011 => 111011000110
[1,1,0,1,0,0,1,1,0,0,1,0] => 110100110010 => 100100010011 => 110001000110
[1,1,0,1,0,0,1,1,0,1,0,0] => 110100110100 => 101100010011 => 111001000110
[1,1,0,1,0,0,1,1,1,0,0,0] => 110100111000 => 101000010011 => 111101000110
[1,1,0,1,0,1,0,0,1,0,1,0] => 110101001010 => 100110110011 => 110011100110
[1,1,0,1,0,1,0,0,1,1,0,0] => 110101001100 => 101110110011 => 111011100110
[1,1,0,1,0,1,0,1,0,0,1,0] => 110101010010 => 100100110011 => 110001100110
[1,1,0,1,0,1,0,1,0,1,0,0] => 110101010100 => 101100110011 => 111001100110
[1,1,0,1,0,1,0,1,1,0,0,0] => 110101011000 => 101000110011 => 111101100110
[1,1,0,1,0,1,1,0,0,0,1,0] => 110101100010 => 100101110011 => 110000100110
[1,1,0,1,0,1,1,0,0,1,0,0] => 110101100100 => 101101110011 => 111000100110
[1,1,0,1,0,1,1,0,1,0,0,0] => 110101101000 => 101001110011 => 111100100110
[1,1,0,1,0,1,1,1,0,0,0,0] => 110101110000 => 101011110011 => 111110100110
[1,1,0,1,1,0,0,0,1,0,1,0] => 110110001010 => 100110100011 => 110011110110
[1,1,0,1,1,0,0,0,1,1,0,0] => 110110001100 => 101110100011 => 111011110110
[1,1,0,1,1,0,0,1,0,0,1,0] => 110110010010 => 100100100011 => 110001110110
[1,1,0,1,1,0,0,1,0,1,0,0] => 110110010100 => 101100100011 => 111001110110
[1,1,0,1,1,0,0,1,1,0,0,0] => 110110011000 => 101000100011 => 111101110110
[1,1,0,1,1,0,1,0,0,0,1,0] => 110110100010 => 100101100011 => 110000110110
[1,1,0,1,1,0,1,0,0,1,0,0] => 110110100100 => 101101100011 => 111000110110
[1,1,0,1,1,0,1,0,1,0,0,0] => 110110101000 => 101001100011 => 111100110110
[1,1,0,1,1,0,1,1,0,0,0,0] => 110110110000 => 101011100011 => 111110110110
[1,1,0,1,1,1,0,0,0,0,1,0] => 110111000010 => 100101000011 => 110000010110
[1,1,0,1,1,1,0,0,0,1,0,0] => 110111000100 => 101101000011 => 111000010110
[1,1,0,1,1,1,0,0,1,0,0,0] => 110111001000 => 101001000011 => 111100010110
[1,1,0,1,1,1,0,1,0,0,0,0] => 110111010000 => 101011000011 => 111110010110
[1,1,0,1,1,1,1,0,0,0,0,0] => 110111100000 => 101010000011 => 111111010110
[1,1,1,0,0,0,1,0,1,0,1,0] => 111000101010 => 100110010111 => 110011000010
[1,1,1,0,0,0,1,0,1,1,0,0] => 111000101100 => 101110010111 => 111011000010
[1,1,1,0,0,0,1,1,0,0,1,0] => 111000110010 => 100100010111 => 110001000010
[1,1,1,0,0,0,1,1,0,1,0,0] => 111000110100 => 101100010111 => 111001000010
[1,1,1,0,0,0,1,1,1,0,0,0] => 111000111000 => 101000010111 => 111101000010
[1,1,1,0,0,1,0,0,1,0,1,0] => 111001001010 => 100110110111 => 110011100010
[1,1,1,0,0,1,0,0,1,1,0,0] => 111001001100 => 101110110111 => 111011100010
[1,1,1,0,0,1,0,1,0,0,1,0] => 111001010010 => 100100110111 => 110001100010
[1,1,1,0,0,1,0,1,0,1,0,0] => 111001010100 => 101100110111 => 111001100010
[1,1,1,0,0,1,0,1,1,0,0,0] => 111001011000 => 101000110111 => 111101100010
[1,1,1,0,0,1,1,0,0,0,1,0] => 111001100010 => 100101110111 => 110000100010
[1,1,1,0,0,1,1,0,0,1,0,0] => 111001100100 => 101101110111 => 111000100010
[1,1,1,0,0,1,1,0,1,0,0,0] => 111001101000 => 101001110111 => 111100100010
[1,1,1,0,0,1,1,1,0,0,0,0] => 111001110000 => 101011110111 => 111110100010
[1,1,1,0,1,0,0,0,1,0,1,0] => 111010001010 => 100110100111 => 110011110010
[1,1,1,0,1,0,0,0,1,1,0,0] => 111010001100 => 101110100111 => 111011110010
[1,1,1,0,1,0,0,1,0,0,1,0] => 111010010010 => 100100100111 => 110001110010
[1,1,1,0,1,0,0,1,0,1,0,0] => 111010010100 => 101100100111 => 111001110010
[1,1,1,0,1,0,0,1,1,0,0,0] => 111010011000 => 101000100111 => 111101110010
[1,1,1,0,1,0,1,0,0,0,1,0] => 111010100010 => 100101100111 => 110000110010
[1,1,1,0,1,0,1,0,0,1,0,0] => 111010100100 => 101101100111 => 111000110010
[1,1,1,0,1,0,1,0,1,0,0,0] => 111010101000 => 101001100111 => 111100110010
[1,1,1,0,1,0,1,1,0,0,0,0] => 111010110000 => 101011100111 => 111110110010
[1,1,1,0,1,1,0,0,0,0,1,0] => 111011000010 => 100101000111 => 110000010010
[1,1,1,0,1,1,0,0,0,1,0,0] => 111011000100 => 101101000111 => 111000010010
[1,1,1,0,1,1,0,0,1,0,0,0] => 111011001000 => 101001000111 => 111100010010
[1,1,1,0,1,1,0,1,0,0,0,0] => 111011010000 => 101011000111 => 111110010010
[1,1,1,0,1,1,1,0,0,0,0,0] => 111011100000 => 101010000111 => 111111010010
[1,1,1,1,0,0,0,0,1,0,1,0] => 111100001010 => 100110101111 => 110011111010
[1,1,1,1,0,0,0,0,1,1,0,0] => 111100001100 => 101110101111 => 111011111010
[1,1,1,1,0,0,0,1,0,0,1,0] => 111100010010 => 100100101111 => 110001111010
[1,1,1,1,0,0,0,1,0,1,0,0] => 111100010100 => 101100101111 => 111001111010
[1,1,1,1,0,0,0,1,1,0,0,0] => 111100011000 => 101000101111 => 111101111010
[1,1,1,1,0,0,1,0,0,0,1,0] => 111100100010 => 100101101111 => 110000111010
[1,1,1,1,0,0,1,0,0,1,0,0] => 111100100100 => 101101101111 => 111000111010
[1,1,1,1,0,0,1,0,1,0,0,0] => 111100101000 => 101001101111 => 111100111010
[1,1,1,1,0,0,1,1,0,0,0,0] => 111100110000 => 101011101111 => 111110111010
[1,1,1,1,0,1,0,0,0,0,1,0] => 111101000010 => 100101001111 => 110000011010
[1,1,1,1,0,1,0,0,0,1,0,0] => 111101000100 => 101101001111 => 111000011010
[1,1,1,1,0,1,0,0,1,0,0,0] => 111101001000 => 101001001111 => 111100011010
[1,1,1,1,0,1,0,1,0,0,0,0] => 111101010000 => 101011001111 => 111110011010
[1,1,1,1,0,1,1,0,0,0,0,0] => 111101100000 => 101010001111 => 111111011010
[1,1,1,1,1,0,0,0,0,0,1,0] => 111110000010 => 100101011111 => 110000001010
[1,1,1,1,1,0,0,0,0,1,0,0] => 111110000100 => 101101011111 => 111000001010
[1,1,1,1,1,0,0,0,1,0,0,0] => 111110001000 => 101001011111 => 111100001010
[1,1,1,1,1,0,0,1,0,0,0,0] => 111110010000 => 101011011111 => 111110001010
[1,1,1,1,1,0,1,0,0,0,0,0] => 111110100000 => 101010011111 => 111111001010
[1,1,1,1,1,1,0,0,0,0,0,0] => 111111000000 => 101010111111 => 111111101010
[1,0,1,1,1,0,1,1,1,0,0,0,0,0] => 10111011100000 => 01010111100001 => 00000010110100
[1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 11010101010100 => 01001100110011 => 00011001100110
[1,1,0,1,0,1,1,0,1,1,0,0,0,0] => 11010110110000 => 01010001110011 => 00000100100110
[1,1,1,0,1,1,0,0,0,1,0,1,0,0] => 11101100010100 => 01001101000111 => 00011000010010
[1,1,1,0,1,1,0,1,0,1,0,0,0,0] => 11101101010000 => 01010011000111 => 00000110010010
[1,1,1,1,0,0,0,1,1,1,0,0,0,0] => 11110001110000 => 01010000101111 => 00000101111010
[1,1,1,1,0,1,0,0,1,1,0,0,0,0] => 11110100110000 => 01010001001111 => 00000100011010
[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => 11110111000000 => 01010100001111 => 00000001011010
[] => => =>
Map
to binary word
Description
Return the Dyck word as binary word.
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.