Processing math: 85%

Identifier
Values
[1,0] => 10 => 01 => 10 => 0
[1,0,1,0] => 1010 => 0101 => 1010 => 0
[1,1,0,0] => 1100 => 0011 => 1001 => 1
[1,0,1,0,1,0] => 101010 => 010101 => 101010 => 0
[1,0,1,1,0,0] => 101100 => 010011 => 101001 => 1
[1,1,0,0,1,0] => 110010 => 001101 => 100110 => 1
[1,1,0,1,0,0] => 110100 => 001011 => 100101 => 1
[1,1,1,0,0,0] => 111000 => 000111 => 100011 => 2
[1,0,1,0,1,0,1,0] => 10101010 => 01010101 => 10101010 => 0
[1,0,1,0,1,1,0,0] => 10101100 => 01010011 => 10101001 => 1
[1,0,1,1,0,0,1,0] => 10110010 => 01001101 => 10100110 => 1
[1,0,1,1,0,1,0,0] => 10110100 => 01001011 => 10100101 => 1
[1,0,1,1,1,0,0,0] => 10111000 => 01000111 => 10100011 => 2
[1,1,0,0,1,0,1,0] => 11001010 => 00110101 => 10011010 => 1
[1,1,0,0,1,1,0,0] => 11001100 => 00110011 => 10011001 => 1
[1,1,0,1,0,0,1,0] => 11010010 => 00101101 => 10010110 => 1
[1,1,0,1,0,1,0,0] => 11010100 => 00101011 => 10010101 => 1
[1,1,0,1,1,0,0,0] => 11011000 => 00100111 => 10010011 => 2
[1,1,1,0,0,0,1,0] => 11100010 => 00011101 => 10001110 => 2
[1,1,1,0,0,1,0,0] => 11100100 => 00011011 => 10001101 => 2
[1,1,1,0,1,0,0,0] => 11101000 => 00010111 => 10001011 => 2
[1,1,1,1,0,0,0,0] => 11110000 => 00001111 => 10000111 => 3
[1,0,1,0,1,0,1,0,1,0] => 1010101010 => 0101010101 => 1010101010 => 0
[1,0,1,0,1,0,1,1,0,0] => 1010101100 => 0101010011 => 1010101001 => 1
[1,0,1,0,1,1,0,0,1,0] => 1010110010 => 0101001101 => 1010100110 => 1
[1,0,1,0,1,1,0,1,0,0] => 1010110100 => 0101001011 => 1010100101 => 1
[1,0,1,1,0,0,1,0,1,0] => 1011001010 => 0100110101 => 1010011010 => 1
[1,0,1,1,0,0,1,1,0,0] => 1011001100 => 0100110011 => 1010011001 => 1
[1,0,1,1,0,1,0,0,1,0] => 1011010010 => 0100101101 => 1010010110 => 1
[1,0,1,1,0,1,0,1,0,0] => 1011010100 => 0100101011 => 1010010101 => 1
[1,0,1,1,1,0,0,0,1,0] => 1011100010 => 0100011101 => 1010001110 => 2
[1,1,0,0,1,0,1,0,1,0] => 1100101010 => 0011010101 => 1001101010 => 1
[1,1,0,0,1,0,1,1,0,0] => 1100101100 => 0011010011 => 1001101001 => 1
[1,1,0,0,1,1,0,0,1,0] => 1100110010 => 0011001101 => 1001100110 => 1
[1,1,0,0,1,1,0,1,0,0] => 1100110100 => 0011001011 => 1001100101 => 1
[1,1,0,1,0,0,1,0,1,0] => 1101001010 => 0010110101 => 1001011010 => 1
[1,1,0,1,0,0,1,1,0,0] => 1101001100 => 0010110011 => 1001011001 => 1
[1,1,0,1,0,1,0,0,1,0] => 1101010010 => 0010101101 => 1001010110 => 1
[1,1,0,1,0,1,0,1,0,0] => 1101010100 => 0010101011 => 1001010101 => 1
[1,1,0,1,1,0,0,0,1,0] => 1101100010 => 0010011101 => 1001001110 => 2
[1,1,1,0,0,0,1,0,1,0] => 1110001010 => 0001110101 => 1000111010 => 2
[1,1,1,0,0,1,0,0,1,0] => 1110010010 => 0001101101 => 1000110110 => 2
[1,1,1,0,1,0,0,0,1,0] => 1110100010 => 0001011101 => 1000101110 => 2
[1,1,1,1,0,0,0,0,1,0] => 1111000010 => 0000111101 => 1000011110 => 3
[1,0,1,0,1,0,1,0,1,0,1,0] => 101010101010 => 010101010101 => 101010101010 => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => 101010101100 => 010101010011 => 101010101001 => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => 101010110100 => 010101001011 => 101010100101 => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => 101011001100 => 010100110011 => 101010011001 => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => 101011010100 => 010100101011 => 101010010101 => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => 101100101100 => 010011010011 => 101001101001 => 1
[1,0,1,1,0,0,1,1,0,1,0,0] => 101100110100 => 010011001011 => 101001100101 => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => 101101001100 => 010010110011 => 101001011001 => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => 101101010100 => 010010101011 => 101001010101 => 1
[1,1,0,0,1,0,1,0,1,1,0,0] => 110010101100 => 001101010011 => 100110101001 => 1
[1,1,0,0,1,0,1,1,0,1,0,0] => 110010110100 => 001101001011 => 100110100101 => 1
[1,1,0,0,1,1,0,0,1,1,0,0] => 110011001100 => 001100110011 => 100110011001 => 1
[1,1,0,0,1,1,0,1,0,1,0,0] => 110011010100 => 001100101011 => 100110010101 => 1
[1,1,0,1,0,0,1,0,1,1,0,0] => 110100101100 => 001011010011 => 100101101001 => 1
[1,1,0,1,0,0,1,1,0,1,0,0] => 110100110100 => 001011001011 => 100101100101 => 1
[1,1,0,1,0,1,0,0,1,1,0,0] => 110101001100 => 001010110011 => 100101011001 => 1
[1,1,0,1,0,1,0,1,0,1,0,0] => 110101010100 => 001010101011 => 100101010101 => 1
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Description
The depth of the binary word interpreted as a path.
This is the maximal value of the number of zeros minus the number of ones occurring in a prefix of the binary word, see [1, sec.9.1.2].
The number of binary words of length n with depth k is \binom{n}{\lfloor\frac{(n+1) - (-1)^{n-k}(k+1)}{2}\rfloor}, see [2].
Map
to binary word
Description
Return the Dyck word as binary word.
Map
rotate back-to-front
Description
The rotation of a binary word, last letter first.
This is the word obtained by moving the last letter to the beginnig.
Map
complement
Description
Send a binary word to the word obtained by interchanging the two letters.