Binary words edit

# 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 **42 statistics** on **Binary words** 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.

The Grundy value for the game of ruler on a binary word.

The stopping time of the decimal representation of the binary word for the 3x+1 p....

The decimal representation of a binary word with a leading 1.

The number of standard Young tableaux whose descent set is the binary word.

The semilength of the longest Dyck word in the Catalan factorisation of a binary ....

The number of factors in the Catalan decomposition of a binary word.

The depth of the binary word interpreted as a path.

The number of ones minus the number of zeros of a binary word.

The number of critical steps in the Catalan decomposition of a binary word.

The number of internal inversions of a binary word.

The minimal number such that all substrings of this length are unique.

# 6. Maps

We have the following **14 maps** in the database:

to binary word

to binary word

to binary word

Foata bijection

delta morphism

reverse

complement

descent word

connectivity set

descent tops

descent bottoms

descent word

rotate front-to-back

rotate back-to-front

# 7. References

# 8. Sage examples