Processing math: 100%

Identifier
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00272: Binary words —Gray next⟶ Binary words
Images
[1,0] => 10 => 11 => 01
[1,0,1,0] => 1010 => 1111 => 0111
[1,1,0,0] => 1100 => 1001 => 0001
[1,0,1,0,1,0] => 101010 => 111111 => 011111
[1,0,1,1,0,0] => 101100 => 111001 => 011001
[1,1,0,0,1,0] => 110010 => 100111 => 000111
[1,1,0,1,0,0] => 110100 => 100001 => 000001
[1,1,1,0,0,0] => 111000 => 101101 => 001101
[1,0,1,0,1,0,1,0] => 10101010 => 11111111 => 01111111
[1,0,1,0,1,1,0,0] => 10101100 => 11111001 => 01111001
[1,0,1,1,0,0,1,0] => 10110010 => 11100111 => 01100111
[1,0,1,1,0,1,0,0] => 10110100 => 11100001 => 01100001
[1,0,1,1,1,0,0,0] => 10111000 => 11101101 => 01101101
[1,1,0,0,1,0,1,0] => 11001010 => 10011111 => 00011111
[1,1,0,0,1,1,0,0] => 11001100 => 10011001 => 00011001
[1,1,0,1,0,0,1,0] => 11010010 => 10000111 => 00000111
[1,1,0,1,0,1,0,0] => 11010100 => 10000001 => 00000001
[1,1,0,1,1,0,0,0] => 11011000 => 10001101 => 00001101
[1,1,1,0,0,0,1,0] => 11100010 => 10110111 => 00110111
[1,1,1,0,0,1,0,0] => 11100100 => 10110001 => 00110001
[1,1,1,0,1,0,0,0] => 11101000 => 10111101 => 00111101
[1,1,1,1,0,0,0,0] => 11110000 => 10100101 => 00100101
[1,0,1,0,1,0,1,0,1,0] => 1010101010 => 1111111111 => 0111111111
[1,0,1,0,1,0,1,1,0,0] => 1010101100 => 1111111001 => 0111111001
[1,0,1,0,1,1,0,0,1,0] => 1010110010 => 1111100111 => 0111100111
[1,0,1,0,1,1,0,1,0,0] => 1010110100 => 1111100001 => 0111100001
[1,0,1,0,1,1,1,0,0,0] => 1010111000 => 1111101101 => 0111101101
[1,0,1,1,0,0,1,0,1,0] => 1011001010 => 1110011111 => 0110011111
[1,0,1,1,0,0,1,1,0,0] => 1011001100 => 1110011001 => 0110011001
[1,0,1,1,0,1,0,0,1,0] => 1011010010 => 1110000111 => 0110000111
[1,0,1,1,0,1,0,1,0,0] => 1011010100 => 1110000001 => 0110000001
[1,0,1,1,0,1,1,0,0,0] => 1011011000 => 1110001101 => 0110001101
[1,0,1,1,1,0,0,0,1,0] => 1011100010 => 1110110111 => 0110110111
[1,0,1,1,1,0,0,1,0,0] => 1011100100 => 1110110001 => 0110110001
[1,0,1,1,1,0,1,0,0,0] => 1011101000 => 1110111101 => 0110111101
[1,0,1,1,1,1,0,0,0,0] => 1011110000 => 1110100101 => 0110100101
[1,1,0,0,1,0,1,0,1,0] => 1100101010 => 1001111111 => 0001111111
[1,1,0,0,1,0,1,1,0,0] => 1100101100 => 1001111001 => 0001111001
[1,1,0,0,1,1,0,0,1,0] => 1100110010 => 1001100111 => 0001100111
[1,1,0,0,1,1,0,1,0,0] => 1100110100 => 1001100001 => 0001100001
[1,1,0,0,1,1,1,0,0,0] => 1100111000 => 1001101101 => 0001101101
[1,1,0,1,0,0,1,0,1,0] => 1101001010 => 1000011111 => 0000011111
[1,1,0,1,0,0,1,1,0,0] => 1101001100 => 1000011001 => 0000011001
[1,1,0,1,0,1,0,0,1,0] => 1101010010 => 1000000111 => 0000000111
[1,1,0,1,0,1,0,1,0,0] => 1101010100 => 1000000001 => 0000000001
[1,1,0,1,0,1,1,0,0,0] => 1101011000 => 1000001101 => 0000001101
[1,1,0,1,1,0,0,0,1,0] => 1101100010 => 1000110111 => 0000110111
[1,1,0,1,1,0,0,1,0,0] => 1101100100 => 1000110001 => 0000110001
[1,1,0,1,1,0,1,0,0,0] => 1101101000 => 1000111101 => 0000111101
[1,1,0,1,1,1,0,0,0,0] => 1101110000 => 1000100101 => 0000100101
[1,1,1,0,0,0,1,0,1,0] => 1110001010 => 1011011111 => 0011011111
[1,1,1,0,0,0,1,1,0,0] => 1110001100 => 1011011001 => 0011011001
[1,1,1,0,0,1,0,0,1,0] => 1110010010 => 1011000111 => 0011000111
[1,1,1,0,0,1,0,1,0,0] => 1110010100 => 1011000001 => 0011000001
[1,1,1,0,0,1,1,0,0,0] => 1110011000 => 1011001101 => 0011001101
[1,1,1,0,1,0,0,0,1,0] => 1110100010 => 1011110111 => 0011110111
[1,1,1,0,1,0,0,1,0,0] => 1110100100 => 1011110001 => 0011110001
[1,1,1,0,1,0,1,0,0,0] => 1110101000 => 1011111101 => 0011111101
[1,1,1,0,1,1,0,0,0,0] => 1110110000 => 1011100101 => 0011100101
[1,1,1,1,0,0,0,0,1,0] => 1111000010 => 1010010111 => 0010010111
[1,1,1,1,0,0,0,1,0,0] => 1111000100 => 1010010001 => 0010010001
[1,1,1,1,0,0,1,0,0,0] => 1111001000 => 1010011101 => 0010011101
[1,1,1,1,0,1,0,0,0,0] => 1111010000 => 1010000101 => 0010000101
[1,1,1,1,1,0,0,0,0,0] => 1111100000 => 1010110101 => 0010110101
[1,0,1,0,1,0,1,0,1,0,1,0] => 101010101010 => 111111111111 => 011111111111
[1,0,1,0,1,0,1,0,1,1,0,0] => 101010101100 => 111111111001 => 011111111001
[1,0,1,0,1,0,1,1,0,0,1,0] => 101010110010 => 111111100111 => 011111100111
[1,0,1,0,1,0,1,1,0,1,0,0] => 101010110100 => 111111100001 => 011111100001
[1,0,1,0,1,0,1,1,1,0,0,0] => 101010111000 => 111111101101 => 011111101101
[1,0,1,0,1,1,0,0,1,0,1,0] => 101011001010 => 111110011111 => 011110011111
[1,0,1,0,1,1,0,0,1,1,0,0] => 101011001100 => 111110011001 => 011110011001
[1,0,1,0,1,1,0,1,0,0,1,0] => 101011010010 => 111110000111 => 011110000111
[1,0,1,0,1,1,0,1,0,1,0,0] => 101011010100 => 111110000001 => 011110000001
[1,0,1,0,1,1,0,1,1,0,0,0] => 101011011000 => 111110001101 => 011110001101
[1,0,1,0,1,1,1,0,0,0,1,0] => 101011100010 => 111110110111 => 011110110111
[1,0,1,0,1,1,1,0,0,1,0,0] => 101011100100 => 111110110001 => 011110110001
[1,0,1,0,1,1,1,0,1,0,0,0] => 101011101000 => 111110111101 => 011110111101
[1,0,1,0,1,1,1,1,0,0,0,0] => 101011110000 => 111110100101 => 011110100101
[1,0,1,1,0,0,1,0,1,0,1,0] => 101100101010 => 111001111111 => 011001111111
[1,0,1,1,0,0,1,0,1,1,0,0] => 101100101100 => 111001111001 => 011001111001
[1,0,1,1,0,0,1,1,0,0,1,0] => 101100110010 => 111001100111 => 011001100111
[1,0,1,1,0,0,1,1,0,1,0,0] => 101100110100 => 111001100001 => 011001100001
[1,0,1,1,0,0,1,1,1,0,0,0] => 101100111000 => 111001101101 => 011001101101
[1,0,1,1,0,1,0,0,1,0,1,0] => 101101001010 => 111000011111 => 011000011111
[1,0,1,1,0,1,0,0,1,1,0,0] => 101101001100 => 111000011001 => 011000011001
[1,0,1,1,0,1,0,1,0,0,1,0] => 101101010010 => 111000000111 => 011000000111
[1,0,1,1,0,1,0,1,0,1,0,0] => 101101010100 => 111000000001 => 011000000001
[1,0,1,1,0,1,0,1,1,0,0,0] => 101101011000 => 111000001101 => 011000001101
[1,0,1,1,0,1,1,0,0,0,1,0] => 101101100010 => 111000110111 => 011000110111
[1,0,1,1,0,1,1,0,0,1,0,0] => 101101100100 => 111000110001 => 011000110001
[1,0,1,1,0,1,1,0,1,0,0,0] => 101101101000 => 111000111101 => 011000111101
[1,0,1,1,0,1,1,1,0,0,0,0] => 101101110000 => 111000100101 => 011000100101
[1,0,1,1,1,0,0,0,1,0,1,0] => 101110001010 => 111011011111 => 011011011111
[1,0,1,1,1,0,0,0,1,1,0,0] => 101110001100 => 111011011001 => 011011011001
[1,0,1,1,1,0,0,1,0,0,1,0] => 101110010010 => 111011000111 => 011011000111
[1,0,1,1,1,0,0,1,0,1,0,0] => 101110010100 => 111011000001 => 011011000001
[1,0,1,1,1,0,0,1,1,0,0,0] => 101110011000 => 111011001101 => 011011001101
[1,0,1,1,1,0,1,0,0,0,1,0] => 101110100010 => 111011110111 => 011011110111
[1,0,1,1,1,0,1,0,0,1,0,0] => 101110100100 => 111011110001 => 011011110001
[1,0,1,1,1,0,1,0,1,0,0,0] => 101110101000 => 111011111101 => 011011111101
[1,0,1,1,1,0,1,1,0,0,0,0] => 101110110000 => 111011100101 => 011011100101
>>> Load all 197 entries. <<<
[1,0,1,1,1,1,0,0,0,0,1,0] => 101111000010 => 111010010111 => 011010010111
[1,0,1,1,1,1,0,0,0,1,0,0] => 101111000100 => 111010010001 => 011010010001
[1,0,1,1,1,1,0,0,1,0,0,0] => 101111001000 => 111010011101 => 011010011101
[1,0,1,1,1,1,0,1,0,0,0,0] => 101111010000 => 111010000101 => 011010000101
[1,0,1,1,1,1,1,0,0,0,0,0] => 101111100000 => 111010110101 => 011010110101
[1,1,0,0,1,0,1,0,1,0,1,0] => 110010101010 => 100111111111 => 000111111111
[1,1,0,0,1,0,1,0,1,1,0,0] => 110010101100 => 100111111001 => 000111111001
[1,1,0,0,1,0,1,1,0,0,1,0] => 110010110010 => 100111100111 => 000111100111
[1,1,0,0,1,0,1,1,0,1,0,0] => 110010110100 => 100111100001 => 000111100001
[1,1,0,0,1,0,1,1,1,0,0,0] => 110010111000 => 100111101101 => 000111101101
[1,1,0,0,1,1,0,0,1,0,1,0] => 110011001010 => 100110011111 => 000110011111
[1,1,0,0,1,1,0,0,1,1,0,0] => 110011001100 => 100110011001 => 000110011001
[1,1,0,0,1,1,0,1,0,0,1,0] => 110011010010 => 100110000111 => 000110000111
[1,1,0,0,1,1,0,1,0,1,0,0] => 110011010100 => 100110000001 => 000110000001
[1,1,0,0,1,1,0,1,1,0,0,0] => 110011011000 => 100110001101 => 000110001101
[1,1,0,0,1,1,1,0,0,0,1,0] => 110011100010 => 100110110111 => 000110110111
[1,1,0,0,1,1,1,0,0,1,0,0] => 110011100100 => 100110110001 => 000110110001
[1,1,0,0,1,1,1,0,1,0,0,0] => 110011101000 => 100110111101 => 000110111101
[1,1,0,0,1,1,1,1,0,0,0,0] => 110011110000 => 100110100101 => 000110100101
[1,1,0,1,0,0,1,0,1,0,1,0] => 110100101010 => 100001111111 => 000001111111
[1,1,0,1,0,0,1,0,1,1,0,0] => 110100101100 => 100001111001 => 000001111001
[1,1,0,1,0,0,1,1,0,0,1,0] => 110100110010 => 100001100111 => 000001100111
[1,1,0,1,0,0,1,1,0,1,0,0] => 110100110100 => 100001100001 => 000001100001
[1,1,0,1,0,0,1,1,1,0,0,0] => 110100111000 => 100001101101 => 000001101101
[1,1,0,1,0,1,0,0,1,0,1,0] => 110101001010 => 100000011111 => 000000011111
[1,1,0,1,0,1,0,0,1,1,0,0] => 110101001100 => 100000011001 => 000000011001
[1,1,0,1,0,1,0,1,0,0,1,0] => 110101010010 => 100000000111 => 000000000111
[1,1,0,1,0,1,0,1,0,1,0,0] => 110101010100 => 100000000001 => 000000000001
[1,1,0,1,0,1,0,1,1,0,0,0] => 110101011000 => 100000001101 => 000000001101
[1,1,0,1,0,1,1,0,0,0,1,0] => 110101100010 => 100000110111 => 000000110111
[1,1,0,1,0,1,1,0,0,1,0,0] => 110101100100 => 100000110001 => 000000110001
[1,1,0,1,0,1,1,0,1,0,0,0] => 110101101000 => 100000111101 => 000000111101
[1,1,0,1,0,1,1,1,0,0,0,0] => 110101110000 => 100000100101 => 000000100101
[1,1,0,1,1,0,0,0,1,0,1,0] => 110110001010 => 100011011111 => 000011011111
[1,1,0,1,1,0,0,0,1,1,0,0] => 110110001100 => 100011011001 => 000011011001
[1,1,0,1,1,0,0,1,0,0,1,0] => 110110010010 => 100011000111 => 000011000111
[1,1,0,1,1,0,0,1,0,1,0,0] => 110110010100 => 100011000001 => 000011000001
[1,1,0,1,1,0,0,1,1,0,0,0] => 110110011000 => 100011001101 => 000011001101
[1,1,0,1,1,0,1,0,0,0,1,0] => 110110100010 => 100011110111 => 000011110111
[1,1,0,1,1,0,1,0,0,1,0,0] => 110110100100 => 100011110001 => 000011110001
[1,1,0,1,1,0,1,0,1,0,0,0] => 110110101000 => 100011111101 => 000011111101
[1,1,0,1,1,0,1,1,0,0,0,0] => 110110110000 => 100011100101 => 000011100101
[1,1,0,1,1,1,0,0,0,0,1,0] => 110111000010 => 100010010111 => 000010010111
[1,1,0,1,1,1,0,0,0,1,0,0] => 110111000100 => 100010010001 => 000010010001
[1,1,0,1,1,1,0,0,1,0,0,0] => 110111001000 => 100010011101 => 000010011101
[1,1,0,1,1,1,0,1,0,0,0,0] => 110111010000 => 100010000101 => 000010000101
[1,1,0,1,1,1,1,0,0,0,0,0] => 110111100000 => 100010110101 => 000010110101
[1,1,1,0,0,0,1,0,1,0,1,0] => 111000101010 => 101101111111 => 001101111111
[1,1,1,0,0,0,1,0,1,1,0,0] => 111000101100 => 101101111001 => 001101111001
[1,1,1,0,0,0,1,1,0,0,1,0] => 111000110010 => 101101100111 => 001101100111
[1,1,1,0,0,0,1,1,0,1,0,0] => 111000110100 => 101101100001 => 001101100001
[1,1,1,0,0,0,1,1,1,0,0,0] => 111000111000 => 101101101101 => 001101101101
[1,1,1,0,0,1,0,0,1,0,1,0] => 111001001010 => 101100011111 => 001100011111
[1,1,1,0,0,1,0,0,1,1,0,0] => 111001001100 => 101100011001 => 001100011001
[1,1,1,0,0,1,0,1,0,0,1,0] => 111001010010 => 101100000111 => 001100000111
[1,1,1,0,0,1,0,1,0,1,0,0] => 111001010100 => 101100000001 => 001100000001
[1,1,1,0,0,1,0,1,1,0,0,0] => 111001011000 => 101100001101 => 001100001101
[1,1,1,0,0,1,1,0,0,0,1,0] => 111001100010 => 101100110111 => 001100110111
[1,1,1,0,0,1,1,0,0,1,0,0] => 111001100100 => 101100110001 => 001100110001
[1,1,1,0,0,1,1,0,1,0,0,0] => 111001101000 => 101100111101 => 001100111101
[1,1,1,0,0,1,1,1,0,0,0,0] => 111001110000 => 101100100101 => 001100100101
[1,1,1,0,1,0,0,0,1,0,1,0] => 111010001010 => 101111011111 => 001111011111
[1,1,1,0,1,0,0,0,1,1,0,0] => 111010001100 => 101111011001 => 001111011001
[1,1,1,0,1,0,0,1,0,0,1,0] => 111010010010 => 101111000111 => 001111000111
[1,1,1,0,1,0,0,1,0,1,0,0] => 111010010100 => 101111000001 => 001111000001
[1,1,1,0,1,0,0,1,1,0,0,0] => 111010011000 => 101111001101 => 001111001101
[1,1,1,0,1,0,1,0,0,0,1,0] => 111010100010 => 101111110111 => 001111110111
[1,1,1,0,1,0,1,0,0,1,0,0] => 111010100100 => 101111110001 => 001111110001
[1,1,1,0,1,0,1,0,1,0,0,0] => 111010101000 => 101111111101 => 001111111101
[1,1,1,0,1,0,1,1,0,0,0,0] => 111010110000 => 101111100101 => 001111100101
[1,1,1,0,1,1,0,0,0,0,1,0] => 111011000010 => 101110010111 => 001110010111
[1,1,1,0,1,1,0,0,0,1,0,0] => 111011000100 => 101110010001 => 001110010001
[1,1,1,0,1,1,0,0,1,0,0,0] => 111011001000 => 101110011101 => 001110011101
[1,1,1,0,1,1,0,1,0,0,0,0] => 111011010000 => 101110000101 => 001110000101
[1,1,1,0,1,1,1,0,0,0,0,0] => 111011100000 => 101110110101 => 001110110101
[1,1,1,1,0,0,0,0,1,0,1,0] => 111100001010 => 101001011111 => 001001011111
[1,1,1,1,0,0,0,0,1,1,0,0] => 111100001100 => 101001011001 => 001001011001
[1,1,1,1,0,0,0,1,0,0,1,0] => 111100010010 => 101001000111 => 001001000111
[1,1,1,1,0,0,0,1,0,1,0,0] => 111100010100 => 101001000001 => 001001000001
[1,1,1,1,0,0,0,1,1,0,0,0] => 111100011000 => 101001001101 => 001001001101
[1,1,1,1,0,0,1,0,0,0,1,0] => 111100100010 => 101001110111 => 001001110111
[1,1,1,1,0,0,1,0,0,1,0,0] => 111100100100 => 101001110001 => 001001110001
[1,1,1,1,0,0,1,0,1,0,0,0] => 111100101000 => 101001111101 => 001001111101
[1,1,1,1,0,0,1,1,0,0,0,0] => 111100110000 => 101001100101 => 001001100101
[1,1,1,1,0,1,0,0,0,0,1,0] => 111101000010 => 101000010111 => 001000010111
[1,1,1,1,0,1,0,0,0,1,0,0] => 111101000100 => 101000010001 => 001000010001
[1,1,1,1,0,1,0,0,1,0,0,0] => 111101001000 => 101000011101 => 001000011101
[1,1,1,1,0,1,0,1,0,0,0,0] => 111101010000 => 101000000101 => 001000000101
[1,1,1,1,0,1,1,0,0,0,0,0] => 111101100000 => 101000110101 => 001000110101
[1,1,1,1,1,0,0,0,0,0,1,0] => 111110000010 => 101011010111 => 001011010111
[1,1,1,1,1,0,0,0,0,1,0,0] => 111110000100 => 101011010001 => 001011010001
[1,1,1,1,1,0,0,0,1,0,0,0] => 111110001000 => 101011011101 => 001011011101
[1,1,1,1,1,0,0,1,0,0,0,0] => 111110010000 => 101011000101 => 001011000101
[1,1,1,1,1,0,1,0,0,0,0,0] => 111110100000 => 101011110101 => 001011110101
[1,1,1,1,1,1,0,0,0,0,0,0] => 111111000000 => 101010010101 => 001010010101
[] => => =>
Map
to binary word
Description
Return the Dyck word as binary word.
Map
alternating inverse
Description
Sends a binary word w1⋯wm to the binary word v1⋯vm with vi=wi if i is odd and vi=1−wi if i is even.
This map is used in [1], see Definitions 3.2 and 5.1.
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.