Identifier
Values
[.,.] => [1,0] => 10 => 11 => 0
[.,[.,.]] => [1,0,1,0] => 1010 => 1101 => 0
[[.,.],.] => [1,1,0,0] => 1100 => 0111 => 0
[.,[.,[.,.]]] => [1,0,1,0,1,0] => 101010 => 110101 => 0
[.,[[.,.],.]] => [1,0,1,1,0,0] => 101100 => 110011 => 0
[[.,.],[.,.]] => [1,1,0,0,1,0] => 110010 => 011101 => 0
[[.,[.,.]],.] => [1,1,0,1,0,0] => 110100 => 111001 => 0
[[[.,.],.],.] => [1,1,1,0,0,0] => 111000 => 001111 => 0
[.,[.,[.,[.,.]]]] => [1,0,1,0,1,0,1,0] => 10101010 => 11010101 => 0
[.,[.,[[.,.],.]]] => [1,0,1,0,1,1,0,0] => 10101100 => 11010011 => 1
[.,[[.,.],[.,.]]] => [1,0,1,1,0,0,1,0] => 10110010 => 11001101 => 0
[.,[[.,[.,.]],.]] => [1,0,1,1,0,1,0,0] => 10110100 => 11011001 => 0
[.,[[[.,.],.],.]] => [1,0,1,1,1,0,0,0] => 10111000 => 11000111 => 0
[[.,.],[.,[.,.]]] => [1,1,0,0,1,0,1,0] => 11001010 => 01110101 => 0
[[.,.],[[.,.],.]] => [1,1,0,0,1,1,0,0] => 11001100 => 01110011 => 0
[[.,[.,.]],[.,.]] => [1,1,0,1,0,0,1,0] => 11010010 => 11100101 => 0
[[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => 11100010 => 00111101 => 0
[[.,[.,[.,.]]],.] => [1,1,0,1,0,1,0,0] => 11010100 => 11101001 => 0
[[.,[[.,.],.]],.] => [1,1,0,1,1,0,0,0] => 11011000 => 11100011 => 0
[[[.,.],[.,.]],.] => [1,1,1,0,0,1,0,0] => 11100100 => 01111001 => 0
[[[.,[.,.]],.],.] => [1,1,1,0,1,0,0,0] => 11101000 => 11110001 => 0
[[[[.,.],.],.],.] => [1,1,1,1,0,0,0,0] => 11110000 => 00011111 => 0
[[[[.,.],.],.],[.,.]] => [1,1,1,1,0,0,0,0,1,0] => 1111000010 => 0001111101 => 0
[[[.,.],[.,[.,.]]],.] => [1,1,1,0,0,1,0,1,0,0] => 1110010100 => 0111101001 => 0
[[[[.,.],[.,.]],.],.] => [1,1,1,1,0,0,1,0,0,0] => 1111001000 => 0111110001 => 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
/ 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 Dyck path: up step, left tree, down step, right tree
Description
Return the associated Dyck path, using the bijection 1L0R.
This is given recursively as follows:
  • a leaf is associated to the empty Dyck Word
  • a tree with children $l,r$ is associated with the Dyck path described by 1L0R where $L$ and $R$ are respectively the Dyck words associated with the trees $l$ and $r$.
Map
to binary word
Description
Return the Dyck word as binary word.
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.