Identifier
Values
[1,0] => [1,0] => 10 => 01 => 0
[1,0,1,0] => [1,1,0,0] => 1100 => 1011 => 0
[1,1,0,0] => [1,0,1,0] => 1010 => 1001 => 0
[1,0,1,0,1,0] => [1,1,1,0,0,0] => 111000 => 010111 => 0
[1,0,1,1,0,0] => [1,1,0,0,1,0] => 110010 => 011011 => 0
[1,1,0,0,1,0] => [1,0,1,1,0,0] => 101100 => 010001 => 0
[1,1,0,1,0,0] => [1,1,0,1,0,0] => 110100 => 010011 => 0
[1,1,1,0,0,0] => [1,0,1,0,1,0] => 101010 => 011001 => 0
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 11110000 => 10101111 => 0
[1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 11100010 => 10010111 => 0
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => 10111011 => 0
[1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => 11100100 => 10110111 => 0
[1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => 11001010 => 10011011 => 0
[1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => 10111000 => 10100001 => 0
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => 10110010 => 10010001 => 0
[1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 11011000 => 10100011 => 0
[1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 11101000 => 10100111 => 0
[1,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0] => 11010010 => 10010011 => 0
[1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => 10101100 => 10111001 => 0
[1,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0] => 10110100 => 10110001 => 0
[1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => 11010100 => 10110011 => 0
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 10101010 => 10011001 => 0
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1111100000 => 0101011111 => 0
[1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 1110001100 => 0100010111 => 0
[1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 1111000100 => 0100101111 => 0
[1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 1100101100 => 0100011011 => 0
[1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => 1011110000 => 0101000001 => 0
[1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 1011100010 => 0110100001 => 0
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 1110110000 => 0101000111 => 0
[1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,1,0,0] => 1110100100 => 0100100111 => 0
[1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => 1011011000 => 0101110001 => 0
[1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 1011010010 => 0110110001 => 0
[1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 1101010100 => 0100110011 => 0
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 1010101010 => 0110011001 => 0
[1,1,0,0,1,0,1,1,0,0,1,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => 101110001100 => 101110100001 => 0
[1,1,0,0,1,1,1,0,0,0,1,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => 101100101100 => 101110010001 => 0
[1,1,1,0,0,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,0,0,1,0] => 101101100010 => 100101110001 => 1
[1,1,1,0,0,1,0,1,0,0,1,0] => [1,0,1,1,1,0,1,1,0,0,0,0] => 101110110000 => 101011100001 => 0
[1,1,1,0,0,1,1,0,1,0,0,0] => [1,0,1,1,1,0,1,0,0,1,0,0] => 101110100100 => 101101100001 => 0
[1,1,1,0,0,1,1,1,0,0,0,0] => [1,0,1,1,0,1,0,0,1,0,1,0] => 101101001010 => 100110110001 => 0
[1,1,1,1,0,0,1,0,0,1,0,0] => [1,0,1,1,0,1,1,0,1,0,0,0] => 101101101000 => 101001110001 => 0
[1,1,1,1,1,0,0,0,0,1,0,0] => [1,0,1,0,1,0,1,1,0,1,0,0] => 101010110100 => 101100011001 => 0
[1,1,1,1,1,0,0,1,0,0,0,0] => [1,0,1,1,0,1,0,1,0,1,0,0] => 101101010100 => 101100110001 => 0
[1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 101010101010 => 100110011001 => 0
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
Description
The defect of a binary word.
The defect of a finite word $w$ is given by the difference between the maximum possible number and the actual number of palindromic factors contained in $w$. The maximum possible number of palindromic factors in a word $w$ is $|w|+1$.
Map
to binary word
Description
Return the Dyck word as binary word.
Map
Lalanne-Kreweras involution
Description
The Lalanne-Kreweras involution on Dyck paths.
Label the upsteps from left to right and record the labels on the first up step of each double rise. Do the same for the downsteps. Then form the Dyck path whose ascent lengths and descent lengths are the consecutives differences of the labels.
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.