Identifier
Values
[1,0] => [1,0] => 10 => 11 => 0
[1,0,1,0] => [1,1,0,0] => 1100 => 0111 => 0
[1,1,0,0] => [1,0,1,0] => 1010 => 1101 => 0
[1,0,1,0,1,0] => [1,1,0,1,0,0] => 110100 => 111001 => 0
[1,0,1,1,0,0] => [1,1,0,0,1,0] => 110010 => 011101 => 0
[1,1,0,0,1,0] => [1,0,1,1,0,0] => 101100 => 110011 => 0
[1,1,0,1,0,0] => [1,1,1,0,0,0] => 111000 => 001111 => 0
[1,1,1,0,0,0] => [1,0,1,0,1,0] => 101010 => 110101 => 0
[1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 11010100 => 11101001 => 0
[1,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => 11010010 => 11100101 => 0
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => 01110011 => 0
[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 11011000 => 11100011 => 0
[1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => 11001010 => 01110101 => 0
[1,1,0,0,1,0,1,0] => [1,0,1,1,0,1,0,0] => 10110100 => 11011001 => 0
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => 10110010 => 11001101 => 0
[1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 11100100 => 01111001 => 0
[1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,0] => 11110000 => 00011111 => 0
[1,1,0,1,1,0,0,0] => [1,1,1,0,0,0,1,0] => 11100010 => 00111101 => 0
[1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => 10101100 => 11010011 => 1
[1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0] => 10111000 => 11000111 => 0
[1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0] => 11101000 => 11110001 => 0
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 10101010 => 11010101 => 0
[1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 1110010100 => 0111101001 => 0
[1,1,0,1,0,1,1,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => 1111000010 => 0001111101 => 0
[1,1,0,1,1,0,1,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => 1111001000 => 0111110001 => 0
[1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1111100000 => 0000111111 => 0
search for individual values
searching the database for the individual values of this statistic
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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
path rowmotion
Description
Return the rowmotion of the binary word, regarded as a lattice path.
Consider the binary word of length $n$ as a lattice path with $n$ steps, where a 1 corresponds to an up step and a 0 corresponds to a down step.
This map returns the path whose peaks are the valleys of the original path with an up step appended.
Map
to binary word
Description
Return the Dyck word as binary word.
Map
Delest-Viennot
Description
Return the Dyck path corresponding to the parallelogram polyomino obtained by applying Delest-Viennot's bijection.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
The Delest-Viennot bijection $\beta$ returns the parallelogram polyomino, whose column heights are the heights of the peaks of the Dyck path, and the intersection heights between columns are the heights of the valleys of the Dyck path.
This map returns the Dyck path $(\gamma^{(-1)}\circ\beta)(D)$.