Identifier
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
Images
[1,0] => 10 => 11 => 11
[1,0,1,0] => 1010 => 1111 => 1111
[1,1,0,0] => 1100 => 1001 => 1010
[1,0,1,0,1,0] => 101010 => 111111 => 111111
[1,0,1,1,0,0] => 101100 => 111001 => 101011
[1,1,0,0,1,0] => 110010 => 100111 => 111010
[1,1,0,1,0,0] => 110100 => 100001 => 101110
[1,1,1,0,0,0] => 111000 => 101101 => 100100
[1,0,1,0,1,0,1,0] => 10101010 => 11111111 => 11111111
[1,0,1,0,1,1,0,0] => 10101100 => 11111001 => 10101111
[1,0,1,1,0,0,1,0] => 10110010 => 11100111 => 11101011
[1,0,1,1,0,1,0,0] => 10110100 => 11100001 => 10111011
[1,0,1,1,1,0,0,0] => 10111000 => 11101101 => 10010011
[1,1,0,0,1,0,1,0] => 11001010 => 10011111 => 11111010
[1,1,0,0,1,1,0,0] => 11001100 => 10011001 => 10101010
[1,1,0,1,0,0,1,0] => 11010010 => 10000111 => 11101110
[1,1,0,1,0,1,0,0] => 11010100 => 10000001 => 10111110
[1,1,0,1,1,0,0,0] => 11011000 => 10001101 => 10010110
[1,1,1,0,0,0,1,0] => 11100010 => 10110111 => 11100100
[1,1,1,0,0,1,0,0] => 11100100 => 10110001 => 10110100
[1,1,1,0,1,0,0,0] => 11101000 => 10111101 => 10011100
[1,1,1,1,0,0,0,0] => 11110000 => 10100101 => 10001000
[1,0,1,0,1,0,1,0,1,0] => 1010101010 => 1111111111 => 1111111111
[1,0,1,0,1,0,1,1,0,0] => 1010101100 => 1111111001 => 1010111111
[1,0,1,0,1,1,0,0,1,0] => 1010110010 => 1111100111 => 1110101111
[1,0,1,0,1,1,0,1,0,0] => 1010110100 => 1111100001 => 1011101111
[1,0,1,0,1,1,1,0,0,0] => 1010111000 => 1111101101 => 1001001111
[1,0,1,1,0,0,1,0,1,0] => 1011001010 => 1110011111 => 1111101011
[1,0,1,1,0,0,1,1,0,0] => 1011001100 => 1110011001 => 1010101011
[1,0,1,1,0,1,0,0,1,0] => 1011010010 => 1110000111 => 1110111011
[1,0,1,1,0,1,0,1,0,0] => 1011010100 => 1110000001 => 1011111011
[1,0,1,1,0,1,1,0,0,0] => 1011011000 => 1110001101 => 1001011011
[1,0,1,1,1,0,0,0,1,0] => 1011100010 => 1110110111 => 1110010011
[1,0,1,1,1,0,0,1,0,0] => 1011100100 => 1110110001 => 1011010011
[1,0,1,1,1,0,1,0,0,0] => 1011101000 => 1110111101 => 1001110011
[1,0,1,1,1,1,0,0,0,0] => 1011110000 => 1110100101 => 1000100011
[1,1,0,0,1,0,1,0,1,0] => 1100101010 => 1001111111 => 1111111010
[1,1,0,0,1,0,1,1,0,0] => 1100101100 => 1001111001 => 1010111010
[1,1,0,0,1,1,0,0,1,0] => 1100110010 => 1001100111 => 1110101010
[1,1,0,0,1,1,0,1,0,0] => 1100110100 => 1001100001 => 1011101010
[1,1,0,0,1,1,1,0,0,0] => 1100111000 => 1001101101 => 1001001010
[1,1,0,1,0,0,1,0,1,0] => 1101001010 => 1000011111 => 1111101110
[1,1,0,1,0,0,1,1,0,0] => 1101001100 => 1000011001 => 1010101110
[1,1,0,1,0,1,0,0,1,0] => 1101010010 => 1000000111 => 1110111110
[1,1,0,1,0,1,0,1,0,0] => 1101010100 => 1000000001 => 1011111110
[1,1,0,1,0,1,1,0,0,0] => 1101011000 => 1000001101 => 1001011110
[1,1,0,1,1,0,0,0,1,0] => 1101100010 => 1000110111 => 1110010110
[1,1,0,1,1,0,0,1,0,0] => 1101100100 => 1000110001 => 1011010110
[1,1,0,1,1,0,1,0,0,0] => 1101101000 => 1000111101 => 1001110110
[1,1,0,1,1,1,0,0,0,0] => 1101110000 => 1000100101 => 1000100110
[1,1,1,0,0,0,1,0,1,0] => 1110001010 => 1011011111 => 1111100100
[1,1,1,0,0,0,1,1,0,0] => 1110001100 => 1011011001 => 1010100100
[1,1,1,0,0,1,0,0,1,0] => 1110010010 => 1011000111 => 1110110100
[1,1,1,0,0,1,0,1,0,0] => 1110010100 => 1011000001 => 1011110100
[1,1,1,0,0,1,1,0,0,0] => 1110011000 => 1011001101 => 1001010100
[1,1,1,0,1,0,0,0,1,0] => 1110100010 => 1011110111 => 1110011100
[1,1,1,0,1,0,0,1,0,0] => 1110100100 => 1011110001 => 1011011100
[1,1,1,0,1,0,1,0,0,0] => 1110101000 => 1011111101 => 1001111100
[1,1,1,0,1,1,0,0,0,0] => 1110110000 => 1011100101 => 1000101100
[1,1,1,1,0,0,0,0,1,0] => 1111000010 => 1010010111 => 1110001000
[1,1,1,1,0,0,0,1,0,0] => 1111000100 => 1010010001 => 1011001000
[1,1,1,1,0,0,1,0,0,0] => 1111001000 => 1010011101 => 1001101000
[1,1,1,1,0,1,0,0,0,0] => 1111010000 => 1010000101 => 1000111000
[1,1,1,1,1,0,0,0,0,0] => 1111100000 => 1010110101 => 1000010000
[1,0,1,0,1,0,1,0,1,0,1,0] => 101010101010 => 111111111111 => 111111111111
[1,0,1,0,1,0,1,0,1,1,0,0] => 101010101100 => 111111111001 => 101011111111
[1,0,1,0,1,0,1,1,0,0,1,0] => 101010110010 => 111111100111 => 111010111111
[1,0,1,0,1,0,1,1,0,1,0,0] => 101010110100 => 111111100001 => 101110111111
[1,0,1,0,1,0,1,1,1,0,0,0] => 101010111000 => 111111101101 => 100100111111
[1,0,1,0,1,1,0,0,1,0,1,0] => 101011001010 => 111110011111 => 111110101111
[1,0,1,0,1,1,0,0,1,1,0,0] => 101011001100 => 111110011001 => 101010101111
[1,0,1,0,1,1,0,1,0,0,1,0] => 101011010010 => 111110000111 => 111011101111
[1,0,1,0,1,1,0,1,0,1,0,0] => 101011010100 => 111110000001 => 101111101111
[1,0,1,0,1,1,0,1,1,0,0,0] => 101011011000 => 111110001101 => 100101101111
[1,0,1,0,1,1,1,0,0,0,1,0] => 101011100010 => 111110110111 => 111001001111
[1,0,1,0,1,1,1,0,0,1,0,0] => 101011100100 => 111110110001 => 101101001111
[1,0,1,0,1,1,1,0,1,0,0,0] => 101011101000 => 111110111101 => 100111001111
[1,0,1,0,1,1,1,1,0,0,0,0] => 101011110000 => 111110100101 => 100010001111
[1,0,1,1,0,0,1,0,1,0,1,0] => 101100101010 => 111001111111 => 111111101011
[1,0,1,1,0,0,1,0,1,1,0,0] => 101100101100 => 111001111001 => 101011101011
[1,0,1,1,0,0,1,1,0,0,1,0] => 101100110010 => 111001100111 => 111010101011
[1,0,1,1,0,0,1,1,0,1,0,0] => 101100110100 => 111001100001 => 101110101011
[1,0,1,1,0,0,1,1,1,0,0,0] => 101100111000 => 111001101101 => 100100101011
[1,0,1,1,0,1,0,0,1,0,1,0] => 101101001010 => 111000011111 => 111110111011
[1,0,1,1,0,1,0,0,1,1,0,0] => 101101001100 => 111000011001 => 101010111011
[1,0,1,1,0,1,0,1,0,0,1,0] => 101101010010 => 111000000111 => 111011111011
[1,0,1,1,0,1,0,1,0,1,0,0] => 101101010100 => 111000000001 => 101111111011
[1,0,1,1,0,1,0,1,1,0,0,0] => 101101011000 => 111000001101 => 100101111011
[1,0,1,1,0,1,1,0,0,0,1,0] => 101101100010 => 111000110111 => 111001011011
[1,0,1,1,0,1,1,0,0,1,0,0] => 101101100100 => 111000110001 => 101101011011
[1,0,1,1,0,1,1,0,1,0,0,0] => 101101101000 => 111000111101 => 100111011011
[1,0,1,1,0,1,1,1,0,0,0,0] => 101101110000 => 111000100101 => 100010011011
[1,0,1,1,1,0,0,0,1,0,1,0] => 101110001010 => 111011011111 => 111110010011
[1,0,1,1,1,0,0,0,1,1,0,0] => 101110001100 => 111011011001 => 101010010011
[1,0,1,1,1,0,0,1,0,0,1,0] => 101110010010 => 111011000111 => 111011010011
[1,0,1,1,1,0,0,1,0,1,0,0] => 101110010100 => 111011000001 => 101111010011
[1,0,1,1,1,0,0,1,1,0,0,0] => 101110011000 => 111011001101 => 100101010011
[1,0,1,1,1,0,1,0,0,0,1,0] => 101110100010 => 111011110111 => 111001110011
[1,0,1,1,1,0,1,0,0,1,0,0] => 101110100100 => 111011110001 => 101101110011
[1,0,1,1,1,0,1,0,1,0,0,0] => 101110101000 => 111011111101 => 100111110011
[1,0,1,1,1,0,1,1,0,0,0,0] => 101110110000 => 111011100101 => 100010110011
>>> Load all 197 entries. <<<
[1,0,1,1,1,1,0,0,0,0,1,0] => 101111000010 => 111010010111 => 111000100011
[1,0,1,1,1,1,0,0,0,1,0,0] => 101111000100 => 111010010001 => 101100100011
[1,0,1,1,1,1,0,0,1,0,0,0] => 101111001000 => 111010011101 => 100110100011
[1,0,1,1,1,1,0,1,0,0,0,0] => 101111010000 => 111010000101 => 100011100011
[1,0,1,1,1,1,1,0,0,0,0,0] => 101111100000 => 111010110101 => 100001000011
[1,1,0,0,1,0,1,0,1,0,1,0] => 110010101010 => 100111111111 => 111111111010
[1,1,0,0,1,0,1,0,1,1,0,0] => 110010101100 => 100111111001 => 101011111010
[1,1,0,0,1,0,1,1,0,0,1,0] => 110010110010 => 100111100111 => 111010111010
[1,1,0,0,1,0,1,1,0,1,0,0] => 110010110100 => 100111100001 => 101110111010
[1,1,0,0,1,0,1,1,1,0,0,0] => 110010111000 => 100111101101 => 100100111010
[1,1,0,0,1,1,0,0,1,0,1,0] => 110011001010 => 100110011111 => 111110101010
[1,1,0,0,1,1,0,0,1,1,0,0] => 110011001100 => 100110011001 => 101010101010
[1,1,0,0,1,1,0,1,0,0,1,0] => 110011010010 => 100110000111 => 111011101010
[1,1,0,0,1,1,0,1,0,1,0,0] => 110011010100 => 100110000001 => 101111101010
[1,1,0,0,1,1,0,1,1,0,0,0] => 110011011000 => 100110001101 => 100101101010
[1,1,0,0,1,1,1,0,0,0,1,0] => 110011100010 => 100110110111 => 111001001010
[1,1,0,0,1,1,1,0,0,1,0,0] => 110011100100 => 100110110001 => 101101001010
[1,1,0,0,1,1,1,0,1,0,0,0] => 110011101000 => 100110111101 => 100111001010
[1,1,0,0,1,1,1,1,0,0,0,0] => 110011110000 => 100110100101 => 100010001010
[1,1,0,1,0,0,1,0,1,0,1,0] => 110100101010 => 100001111111 => 111111101110
[1,1,0,1,0,0,1,0,1,1,0,0] => 110100101100 => 100001111001 => 101011101110
[1,1,0,1,0,0,1,1,0,0,1,0] => 110100110010 => 100001100111 => 111010101110
[1,1,0,1,0,0,1,1,0,1,0,0] => 110100110100 => 100001100001 => 101110101110
[1,1,0,1,0,0,1,1,1,0,0,0] => 110100111000 => 100001101101 => 100100101110
[1,1,0,1,0,1,0,0,1,0,1,0] => 110101001010 => 100000011111 => 111110111110
[1,1,0,1,0,1,0,0,1,1,0,0] => 110101001100 => 100000011001 => 101010111110
[1,1,0,1,0,1,0,1,0,0,1,0] => 110101010010 => 100000000111 => 111011111110
[1,1,0,1,0,1,0,1,0,1,0,0] => 110101010100 => 100000000001 => 101111111110
[1,1,0,1,0,1,0,1,1,0,0,0] => 110101011000 => 100000001101 => 100101111110
[1,1,0,1,0,1,1,0,0,0,1,0] => 110101100010 => 100000110111 => 111001011110
[1,1,0,1,0,1,1,0,0,1,0,0] => 110101100100 => 100000110001 => 101101011110
[1,1,0,1,0,1,1,0,1,0,0,0] => 110101101000 => 100000111101 => 100111011110
[1,1,0,1,0,1,1,1,0,0,0,0] => 110101110000 => 100000100101 => 100010011110
[1,1,0,1,1,0,0,0,1,0,1,0] => 110110001010 => 100011011111 => 111110010110
[1,1,0,1,1,0,0,0,1,1,0,0] => 110110001100 => 100011011001 => 101010010110
[1,1,0,1,1,0,0,1,0,0,1,0] => 110110010010 => 100011000111 => 111011010110
[1,1,0,1,1,0,0,1,0,1,0,0] => 110110010100 => 100011000001 => 101111010110
[1,1,0,1,1,0,0,1,1,0,0,0] => 110110011000 => 100011001101 => 100101010110
[1,1,0,1,1,0,1,0,0,0,1,0] => 110110100010 => 100011110111 => 111001110110
[1,1,0,1,1,0,1,0,0,1,0,0] => 110110100100 => 100011110001 => 101101110110
[1,1,0,1,1,0,1,0,1,0,0,0] => 110110101000 => 100011111101 => 100111110110
[1,1,0,1,1,0,1,1,0,0,0,0] => 110110110000 => 100011100101 => 100010110110
[1,1,0,1,1,1,0,0,0,0,1,0] => 110111000010 => 100010010111 => 111000100110
[1,1,0,1,1,1,0,0,0,1,0,0] => 110111000100 => 100010010001 => 101100100110
[1,1,0,1,1,1,0,0,1,0,0,0] => 110111001000 => 100010011101 => 100110100110
[1,1,0,1,1,1,0,1,0,0,0,0] => 110111010000 => 100010000101 => 100011100110
[1,1,0,1,1,1,1,0,0,0,0,0] => 110111100000 => 100010110101 => 100001000110
[1,1,1,0,0,0,1,0,1,0,1,0] => 111000101010 => 101101111111 => 111111100100
[1,1,1,0,0,0,1,0,1,1,0,0] => 111000101100 => 101101111001 => 101011100100
[1,1,1,0,0,0,1,1,0,0,1,0] => 111000110010 => 101101100111 => 111010100100
[1,1,1,0,0,0,1,1,0,1,0,0] => 111000110100 => 101101100001 => 101110100100
[1,1,1,0,0,0,1,1,1,0,0,0] => 111000111000 => 101101101101 => 100100100100
[1,1,1,0,0,1,0,0,1,0,1,0] => 111001001010 => 101100011111 => 111110110100
[1,1,1,0,0,1,0,0,1,1,0,0] => 111001001100 => 101100011001 => 101010110100
[1,1,1,0,0,1,0,1,0,0,1,0] => 111001010010 => 101100000111 => 111011110100
[1,1,1,0,0,1,0,1,0,1,0,0] => 111001010100 => 101100000001 => 101111110100
[1,1,1,0,0,1,0,1,1,0,0,0] => 111001011000 => 101100001101 => 100101110100
[1,1,1,0,0,1,1,0,0,0,1,0] => 111001100010 => 101100110111 => 111001010100
[1,1,1,0,0,1,1,0,0,1,0,0] => 111001100100 => 101100110001 => 101101010100
[1,1,1,0,0,1,1,0,1,0,0,0] => 111001101000 => 101100111101 => 100111010100
[1,1,1,0,0,1,1,1,0,0,0,0] => 111001110000 => 101100100101 => 100010010100
[1,1,1,0,1,0,0,0,1,0,1,0] => 111010001010 => 101111011111 => 111110011100
[1,1,1,0,1,0,0,0,1,1,0,0] => 111010001100 => 101111011001 => 101010011100
[1,1,1,0,1,0,0,1,0,0,1,0] => 111010010010 => 101111000111 => 111011011100
[1,1,1,0,1,0,0,1,0,1,0,0] => 111010010100 => 101111000001 => 101111011100
[1,1,1,0,1,0,0,1,1,0,0,0] => 111010011000 => 101111001101 => 100101011100
[1,1,1,0,1,0,1,0,0,0,1,0] => 111010100010 => 101111110111 => 111001111100
[1,1,1,0,1,0,1,0,0,1,0,0] => 111010100100 => 101111110001 => 101101111100
[1,1,1,0,1,0,1,0,1,0,0,0] => 111010101000 => 101111111101 => 100111111100
[1,1,1,0,1,0,1,1,0,0,0,0] => 111010110000 => 101111100101 => 100010111100
[1,1,1,0,1,1,0,0,0,0,1,0] => 111011000010 => 101110010111 => 111000101100
[1,1,1,0,1,1,0,0,0,1,0,0] => 111011000100 => 101110010001 => 101100101100
[1,1,1,0,1,1,0,0,1,0,0,0] => 111011001000 => 101110011101 => 100110101100
[1,1,1,0,1,1,0,1,0,0,0,0] => 111011010000 => 101110000101 => 100011101100
[1,1,1,0,1,1,1,0,0,0,0,0] => 111011100000 => 101110110101 => 100001001100
[1,1,1,1,0,0,0,0,1,0,1,0] => 111100001010 => 101001011111 => 111110001000
[1,1,1,1,0,0,0,0,1,1,0,0] => 111100001100 => 101001011001 => 101010001000
[1,1,1,1,0,0,0,1,0,0,1,0] => 111100010010 => 101001000111 => 111011001000
[1,1,1,1,0,0,0,1,0,1,0,0] => 111100010100 => 101001000001 => 101111001000
[1,1,1,1,0,0,0,1,1,0,0,0] => 111100011000 => 101001001101 => 100101001000
[1,1,1,1,0,0,1,0,0,0,1,0] => 111100100010 => 101001110111 => 111001101000
[1,1,1,1,0,0,1,0,0,1,0,0] => 111100100100 => 101001110001 => 101101101000
[1,1,1,1,0,0,1,0,1,0,0,0] => 111100101000 => 101001111101 => 100111101000
[1,1,1,1,0,0,1,1,0,0,0,0] => 111100110000 => 101001100101 => 100010101000
[1,1,1,1,0,1,0,0,0,0,1,0] => 111101000010 => 101000010111 => 111000111000
[1,1,1,1,0,1,0,0,0,1,0,0] => 111101000100 => 101000010001 => 101100111000
[1,1,1,1,0,1,0,0,1,0,0,0] => 111101001000 => 101000011101 => 100110111000
[1,1,1,1,0,1,0,1,0,0,0,0] => 111101010000 => 101000000101 => 100011111000
[1,1,1,1,0,1,1,0,0,0,0,0] => 111101100000 => 101000110101 => 100001011000
[1,1,1,1,1,0,0,0,0,0,1,0] => 111110000010 => 101011010111 => 111000010000
[1,1,1,1,1,0,0,0,0,1,0,0] => 111110000100 => 101011010001 => 101100010000
[1,1,1,1,1,0,0,0,1,0,0,0] => 111110001000 => 101011011101 => 100110010000
[1,1,1,1,1,0,0,1,0,0,0,0] => 111110010000 => 101011000101 => 100011010000
[1,1,1,1,1,0,1,0,0,0,0,0] => 111110100000 => 101011110101 => 100001110000
[1,1,1,1,1,1,0,0,0,0,0,0] => 111111000000 => 101010010101 => 100000100000
[] => => =>
Map
to binary word
Description
Return the Dyck word as binary word.
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.
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 $0\dots 0$.