Binary words

# 1. Definition

A binary word is a word in the alphabet $\{0,1\}$.

# 2. Examples

The binary words of length $3$ are $$000,001,010,011,100,101,110,111.$$

# 3. Properties

• The number of binary words of length $n$ is $2^n$.

# 4. Remarks

• Binary words can also be considered as monoton lattice paths in $\mathbb{Z_{\geq 0}}^2$ starting at $(0,0)$ and consisting of steps $(1,0)$ and $(0,1)$.

# 5. Statistics

We have the following 31 statistics in the database:

The number of ones in a binary word.
The decimal representation of a binary word.
The major index of a binary word.
The number of descents of a binary word.
The number of ascents of a binary word.
The number of inversions of a binary word.
The number of distinct factors of a binary word.
The length of the border of a binary word.
The length of the symmetric border of a binary word.
The number of leading ones in a binary word.
The position of the first one in a non-zero binary word.
The inversion sum of a binary word.
The non-inversion sum of a binary word.
The number of runs of ones of odd length in a binary word.
The number of runs of ones in a binary word.
The sum of the positions of the ones in a binary word.
The length of the longest run of ones in a binary word.
The number of strictly increasing runs in a binary word.
The number of distinct subsequences in a binary word.
The largest length of a factor maximising the subword complexity.
The number of permutations whose descent word is the given binary word.
The size of the conjugacy class of a binary word.
The minimal period of a binary word.
The exponent of a binary word.
The balance of a binary word.
The defect of a binary word.
The length of the shortest palindromic decomposition of a binary word.
The number of distinct palindromic decompositions of a binary word.
The Grundy value of Welter's game on a binary word.
The number of changes of a binary word.
The Grundy value for the game of Kayles on a binary word.

# 6. Maps

We have the following 4 maps in the database:

Foata bijection
delta morphism
reverse
complement