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
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.