# 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

# 7. References

# 8. Sage examples