# 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 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 non-zero binary word.
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 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