# 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 44 statistics on Binary words in the database:

St000288 Binary words ⟶ ℤ
The number of ones in a binary word.
St000289 Binary words ⟶ ℤ
The decimal representation of a binary word.
St000290 Binary words ⟶ ℤ
The major index of a binary word.
St000291 Binary words ⟶ ℤ
The number of descents of a binary word.
St000292 Binary words ⟶ ℤ
The number of ascents of a binary word.
St000293 Binary words ⟶ ℤ
The number of inversions of a binary word.
St000294 Binary words ⟶ ℤ
The number of distinct factors of a binary word.
St000295 Binary words ⟶ ℤ
The length of the border of a binary word.
St000296 Binary words ⟶ ℤ
The length of the symmetric border of a binary word.
St000297 Binary words ⟶ ℤ
The number of leading ones in a binary word.
St000326 Binary words ⟶ ℤ
The position of the first one in a binary word after appending a 1 at the end.
St000347 Binary words ⟶ ℤ
The inversion sum of a binary word.
St000348 Binary words ⟶ ℤ
The non-inversion sum of a binary word.
St000389 Binary words ⟶ ℤ
The number of runs of ones of odd length in a binary word.
St000390 Binary words ⟶ ℤ
The number of runs of ones in a binary word.
St000391 Binary words ⟶ ℤ
The sum of the positions of the ones in a binary word.
St000392 Binary words ⟶ ℤ
The length of the longest run of ones in a binary word.
St000393 Binary words ⟶ ℤ
The number of strictly increasing runs in a binary word.
St000518 Binary words ⟶ ℤ
The number of distinct subsequences in a binary word.
St000519 Binary words ⟶ ℤ
The largest length of a factor maximising the subword complexity.
St000529 Binary words ⟶ ℤ
The number of permutations whose descent word is the given binary word.
St000543 Binary words ⟶ ℤ
The size of the conjugacy class of a binary word.
St000626 Binary words ⟶ ℤ
The minimal period of a binary word.
St000627 Binary words ⟶ ℤ
The exponent of a binary word.
St000628 Binary words ⟶ ℤ
The balance of a binary word.
St000629 Binary words ⟶ ℤ
The defect of a binary word.
St000630 Binary words ⟶ ℤ
The length of the shortest palindromic decomposition of a binary word.
St000631 Binary words ⟶ ℤ
The number of distinct palindromic decompositions of a binary word.
St000682 Binary words ⟶ ℤ
The Grundy value of Welter's game on a binary word.
St000691 Binary words ⟶ ℤ
The number of changes of a binary word.
St000753 Binary words ⟶ ℤ
The Grundy value for the game of Kayles on a binary word.
St000792 Binary words ⟶ ℤ
The Grundy value for the game of ruler on a binary word.
St000826 Binary words ⟶ ℤ
The stopping time of the decimal representation of the binary word for the 3x+1 p....
St000827 Binary words ⟶ ℤ
The decimal representation of a binary word with a leading 1.
St000847 Binary words ⟶ ℤ
The number of standard Young tableaux whose descent set is the binary word.
St000875 Binary words ⟶ ℤ
The semilength of the longest Dyck word in the Catalan factorisation of a binary ....
St000876 Binary words ⟶ ℤ
The number of factors in the Catalan decomposition of a binary word.
St000877 Binary words ⟶ ℤ
The depth of the binary word interpreted as a path.
St000878 Binary words ⟶ ℤ
The number of ones minus the number of zeros of a binary word.
St000885 Binary words ⟶ ℤ
The number of critical steps in the Catalan decomposition of a binary word.
St000921 Binary words ⟶ ℤ
The number of internal inversions of a binary word.
St000922 Binary words ⟶ ℤ
The minimal number such that all substrings of this length are unique.
St000982 Binary words ⟶ ℤ
The length of the longest constant subword.
St000983 Binary words ⟶ ℤ
The length of the longest alternating subword.

# 6. Maps

We have the following 14 maps from and to Binary words in the database:

Mp00093 Dyck paths ⟶ Binary words
to binary word
Mp00094 Integer compositions ⟶ Binary words
to binary word
Mp00095 Integer partitions ⟶ Binary words
to binary word
Mp00096 Binary words ⟶ Binary words
Foata bijection
Mp00097 Binary words ⟶ Integer compositions
delta morphism
Mp00104 Binary words ⟶ Binary words
reverse
Mp00105 Binary words ⟶ Binary words
complement
Mp00109 Permutations ⟶ Binary words
descent word
Mp00114 Permutations ⟶ Binary words
connectivity set
Mp00130 Permutations ⟶ Binary words
descent tops
Mp00131 Permutations ⟶ Binary words
descent bottoms
Mp00134 Standard tableaux ⟶ Binary words
descent word
Mp00135 Binary words ⟶ Binary words
rotate front-to-back
Mp00136 Binary words ⟶ Binary words
rotate back-to-front