combinatorics on words: an introductionshallit/talks/introcw.pdf ·...
TRANSCRIPT
![Page 1: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/1.jpg)
Combinatorics on Words:
An Introduction
Jeffrey ShallitSchool of Computer Science
University of WaterlooWaterloo, Ontario N2L 3G1
http://www.cs.uwaterloo.ca/~shallit
1 / 201
![Page 2: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/2.jpg)
The Main Themes
periodicity
2 / 201
![Page 3: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/3.jpg)
The Main Themes
periodicity
patterns and pattern avoidance
3 / 201
![Page 4: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/4.jpg)
The Main Themes
periodicity
patterns and pattern avoidance
equations in words
4 / 201
![Page 5: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/5.jpg)
The Main Themes
periodicity
patterns and pattern avoidance
equations in words
infinite words and their properties
5 / 201
![Page 6: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/6.jpg)
Some notation
Σ - a finite nonempty set of symbols - the alphabet
6 / 201
![Page 7: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/7.jpg)
Some notation
Σ - a finite nonempty set of symbols - the alphabet
word - a finite or infinite list of symbols chosen from Σ
7 / 201
![Page 8: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/8.jpg)
Some notation
Σ - a finite nonempty set of symbols - the alphabet
word - a finite or infinite list of symbols chosen from Σ
Σ∗ - set of all finite words
8 / 201
![Page 9: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/9.jpg)
Some notation
Σ - a finite nonempty set of symbols - the alphabet
word - a finite or infinite list of symbols chosen from Σ
Σ∗ - set of all finite words
Σ+ - set of all finite nonempty words
9 / 201
![Page 10: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/10.jpg)
Some notation
Σ - a finite nonempty set of symbols - the alphabet
word - a finite or infinite list of symbols chosen from Σ
Σ∗ - set of all finite words
Σ+ - set of all finite nonempty words
Σω - set of all (right-)infinite words
10 / 201
![Page 11: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/11.jpg)
Some notation
Σ - a finite nonempty set of symbols - the alphabet
word - a finite or infinite list of symbols chosen from Σ
Σ∗ - set of all finite words
Σ+ - set of all finite nonempty words
Σω - set of all (right-)infinite words
Σ∞ = Σ∗ ∪ Σω
11 / 201
![Page 12: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/12.jpg)
More notation
the empty word: ε
12 / 201
![Page 13: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/13.jpg)
More notation
the empty word: ε
w = a1a2 · · · an
13 / 201
![Page 14: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/14.jpg)
More notation
the empty word: ε
w = a1a2 · · · an
w [i ] := ai , w [i ..j ] := aiai+1 · · · aj
14 / 201
![Page 15: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/15.jpg)
More notation
the empty word: ε
w = a1a2 · · · an
w [i ] := ai , w [i ..j ] := aiai+1 · · · aj
w = a0a1a2 · · ·
15 / 201
![Page 16: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/16.jpg)
More notation
the empty word: ε
w = a1a2 · · · an
w [i ] := ai , w [i ..j ] := aiai+1 · · · aj
w = a0a1a2 · · ·
xω = xxx · · ·
16 / 201
![Page 17: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/17.jpg)
More notation
the empty word: ε
w = a1a2 · · · an
w [i ] := ai , w [i ..j ] := aiai+1 · · · aj
w = a0a1a2 · · ·
xω = xxx · · ·
ultimately periodic: z = xyω
17 / 201
![Page 18: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/18.jpg)
More notation
the empty word: ε
w = a1a2 · · · an
w [i ] := ai , w [i ..j ] := aiai+1 · · · aj
w = a0a1a2 · · ·
xω = xxx · · ·
ultimately periodic: z = xyω
Operations: concatenation, raising to powers xn =
n︷ ︸︸ ︷xx · · · x , x0 = ε,
reversal xR
18 / 201
![Page 19: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/19.jpg)
More notation
the empty word: ε
w = a1a2 · · · an
w [i ] := ai , w [i ..j ] := aiai+1 · · · aj
w = a0a1a2 · · ·
xω = xxx · · ·
ultimately periodic: z = xyω
Operations: concatenation, raising to powers xn =
n︷ ︸︸ ︷xx · · · x , x0 = ε,
reversal xR
prefix, suffix, factor, subword
19 / 201
![Page 20: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/20.jpg)
Algebraic framework
semigroup: concatenation is multiplication, associative
20 / 201
![Page 21: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/21.jpg)
Algebraic framework
semigroup: concatenation is multiplication, associative
monoid: semigroup + identity element (ε)
21 / 201
![Page 22: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/22.jpg)
Algebraic framework
semigroup: concatenation is multiplication, associative
monoid: semigroup + identity element (ε)
free monoid: no relations among elements
22 / 201
![Page 23: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/23.jpg)
Algebraic framework
semigroup: concatenation is multiplication, associative
monoid: semigroup + identity element (ε)
free monoid: no relations among elements
group: add inverses of elements a−1
23 / 201
![Page 24: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/24.jpg)
Periodicity
words - fundamentally noncommutative
24 / 201
![Page 25: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/25.jpg)
Periodicity
words - fundamentally noncommutative
casebook 6= bookcase
25 / 201
![Page 26: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/26.jpg)
Periodicity
words - fundamentally noncommutative
casebook 6= bookcase
When do words commute?
26 / 201
![Page 27: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/27.jpg)
Periodicity
words - fundamentally noncommutative
casebook 6= bookcase
When do words commute?Here are two words that “almost” commute:
27 / 201
![Page 28: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/28.jpg)
Periodicity
words - fundamentally noncommutative
casebook 6= bookcase
When do words commute?Here are two words that “almost” commute:
w = 01010 and x = 01011010
28 / 201
![Page 29: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/29.jpg)
Periodicity
words - fundamentally noncommutative
casebook 6= bookcase
When do words commute?Here are two words that “almost” commute:
w = 01010 and x = 01011010
wx = 0101001011010
29 / 201
![Page 30: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/30.jpg)
Periodicity
words - fundamentally noncommutative
casebook 6= bookcase
When do words commute?Here are two words that “almost” commute:
w = 01010 and x = 01011010
wx = 0101001011010
xw = 0101101001010
By the way, this raises the question: can the Hamming distancebetween wx and xw be 1? It can’t; there is a one-line proof.
30 / 201
![Page 31: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/31.jpg)
What are the solutions to x2 = y 3 in words?
- Over N, x2 = y3 iff x is a cube and y is a square
31 / 201
![Page 32: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/32.jpg)
What are the solutions to x2 = y 3 in words?
- Over N, x2 = y3 iff x is a cube and y is a square
- Suggests what solutions of x i = y j will look like over words
32 / 201
![Page 33: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/33.jpg)
What are the solutions to x2 = y 3 in words?
- Over N, x2 = y3 iff x is a cube and y is a square
- Suggests what solutions of x i = y j will look like over words
33 / 201
![Page 34: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/34.jpg)
First Theorem of Lyndon-Schutzenberger
Theorem
Let x , y ∈ Σ+. Then the following three conditions are equivalent:
34 / 201
![Page 35: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/35.jpg)
First Theorem of Lyndon-Schutzenberger
Theorem
Let x , y ∈ Σ+. Then the following three conditions are equivalent:(1) xy = yx;
35 / 201
![Page 36: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/36.jpg)
First Theorem of Lyndon-Schutzenberger
Theorem
Let x , y ∈ Σ+. Then the following three conditions are equivalent:(1) xy = yx;(2) There exist z ∈ Σ+ and integers k, l > 0 such that x = zk andy = z l ;
36 / 201
![Page 37: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/37.jpg)
First Theorem of Lyndon-Schutzenberger
Theorem
Let x , y ∈ Σ+. Then the following three conditions are equivalent:(1) xy = yx;(2) There exist z ∈ Σ+ and integers k, l > 0 such that x = zk andy = z l ;(3) There exist integers i , j > 0 such that x i = y j .
37 / 201
![Page 38: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/38.jpg)
Second Theorem of Lyndon-Schutzenberger
Under what conditions can a string have a nontrivial proper prefixand suffix that are identical?
38 / 201
![Page 39: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/39.jpg)
Second Theorem of Lyndon-Schutzenberger
Under what conditions can a string have a nontrivial proper prefixand suffix that are identical?
Examples in English: reader — begins and ends with r
39 / 201
![Page 40: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/40.jpg)
Second Theorem of Lyndon-Schutzenberger
Under what conditions can a string have a nontrivial proper prefixand suffix that are identical?
Examples in English: reader — begins and ends with r
alfalfa — which begins and ends with alfa
40 / 201
![Page 41: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/41.jpg)
Second Theorem of Lyndon-Schutzenberger
Under what conditions can a string have a nontrivial proper prefixand suffix that are identical?
Examples in English: reader — begins and ends with r
alfalfa — which begins and ends with alfa
The answer is given by the following theorem.
41 / 201
![Page 42: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/42.jpg)
Second Theorem of Lyndon-Schutzenberger
Under what conditions can a string have a nontrivial proper prefixand suffix that are identical?
Examples in English: reader — begins and ends with r
alfalfa — which begins and ends with alfa
The answer is given by the following theorem.
Theorem
Let x , y , z ∈ Σ+. Then xy = yz if and only if there exist u ∈ Σ+,v ∈ Σ∗, and an integer e ≥ 0 such that x = uv, z = vu, andy = (uv)eu = u(vu)e .
42 / 201
![Page 43: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/43.jpg)
Primitive words
We say a word x is a power if it can be expressed as x = y n forsome y 6= ε, n ≥ 2.
43 / 201
![Page 44: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/44.jpg)
Primitive words
We say a word x is a power if it can be expressed as x = y n forsome y 6= ε, n ≥ 2.
A nonpower is called primitive.
44 / 201
![Page 45: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/45.jpg)
Primitive words
We say a word x is a power if it can be expressed as x = y n forsome y 6= ε, n ≥ 2.
A nonpower is called primitive.
Every nonempty word can be written uniquely in the form xk
where x is primitive and k ≥ 1.
45 / 201
![Page 46: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/46.jpg)
Primitive words
We say a word x is a power if it can be expressed as x = y n forsome y 6= ε, n ≥ 2.
A nonpower is called primitive.
Every nonempty word can be written uniquely in the form xk
where x is primitive and k ≥ 1.
Enumeration: there are exactly
∑
d | n
µ(d)kn/d
46 / 201
![Page 47: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/47.jpg)
Primitive words
We say a word x is a power if it can be expressed as x = y n forsome y 6= ε, n ≥ 2.
A nonpower is called primitive.
Every nonempty word can be written uniquely in the form xk
where x is primitive and k ≥ 1.
Enumeration: there are exactly
∑
d | n
µ(d)kn/d
primitive words of length n over a k-letter alphabet. Here µ is theMobius function and the sum is over the divisors of n.
47 / 201
![Page 48: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/48.jpg)
Primitive words
We say a word x is a power if it can be expressed as x = y n forsome y 6= ε, n ≥ 2.
A nonpower is called primitive.
Every nonempty word can be written uniquely in the form xk
where x is primitive and k ≥ 1.
Enumeration: there are exactly
∑
d | n
µ(d)kn/d
primitive words of length n over a k-letter alphabet. Here µ is theMobius function and the sum is over the divisors of n.
Open question: is the set of primitive binary words a CFL?
48 / 201
![Page 49: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/49.jpg)
Conjugates
A word w is a conjugate of a word x if w can be obtained from xby cyclically shifting the letters.
49 / 201
![Page 50: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/50.jpg)
Conjugates
A word w is a conjugate of a word x if w can be obtained from xby cyclically shifting the letters.
For example, the English word enlist is a conjugate of listen.
50 / 201
![Page 51: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/51.jpg)
Conjugates
A word w is a conjugate of a word x if w can be obtained from xby cyclically shifting the letters.
For example, the English word enlist is a conjugate of listen.
A conjugate of a k-th power is a k-th power of a conjugate.
51 / 201
![Page 52: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/52.jpg)
Conjugates
A word w is a conjugate of a word x if w can be obtained from xby cyclically shifting the letters.
For example, the English word enlist is a conjugate of listen.
A conjugate of a k-th power is a k-th power of a conjugate.
Every primitive word has an unbordered conjugate.
52 / 201
![Page 53: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/53.jpg)
Conjugates
A word w is a conjugate of a word x if w can be obtained from xby cyclically shifting the letters.
For example, the English word enlist is a conjugate of listen.
A conjugate of a k-th power is a k-th power of a conjugate.
Every primitive word has an unbordered conjugate.
Lyndon word: lexicographically least among all its conjugates
53 / 201
![Page 54: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/54.jpg)
Conjugates
A word w is a conjugate of a word x if w can be obtained from xby cyclically shifting the letters.
For example, the English word enlist is a conjugate of listen.
A conjugate of a k-th power is a k-th power of a conjugate.
Every primitive word has an unbordered conjugate.
Lyndon word: lexicographically least among all its conjugates
Theorem: Every finite word has a unique factorization as theproduct of Lyndon words w1w2 · · ·wn, where w1 ≥ w2 ≥ w3 · · ·wn.
54 / 201
![Page 55: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/55.jpg)
Fractional powers
We say a word w is a p/q power, for integers p ≥ q ≥ 1, if
w = xbp/qcx ′
for a prefix x ′ of x such that |w |/|x | = p/q.
55 / 201
![Page 56: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/56.jpg)
Fractional powers
We say a word w is a p/q power, for integers p ≥ q ≥ 1, if
w = xbp/qcx ′
for a prefix x ′ of x such that |w |/|x | = p/q.
For example, the French word entente is a 7/3-power, as it canbe written as (ent)2e.
56 / 201
![Page 57: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/57.jpg)
Fractional powers
We say a word w is a p/q power, for integers p ≥ q ≥ 1, if
w = xbp/qcx ′
for a prefix x ′ of x such that |w |/|x | = p/q.
For example, the French word entente is a 7/3-power, as it canbe written as (ent)2e.
The German word schematische is a 3/2 power.
57 / 201
![Page 58: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/58.jpg)
Fractional powers
We say a word w is a p/q power, for integers p ≥ q ≥ 1, if
w = xbp/qcx ′
for a prefix x ′ of x such that |w |/|x | = p/q.
For example, the French word entente is a 7/3-power, as it canbe written as (ent)2e.
The German word schematische is a 3/2 power.
If w = xbp/qcx ′ is a p/q power, then we call x a period of w .Often the word period is used to refer to |x |.
58 / 201
![Page 59: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/59.jpg)
Fractional powers
We say a word w is a p/q power, for integers p ≥ q ≥ 1, if
w = xbp/qcx ′
for a prefix x ′ of x such that |w |/|x | = p/q.
For example, the French word entente is a 7/3-power, as it canbe written as (ent)2e.
The German word schematische is a 3/2 power.
If w = xbp/qcx ′ is a p/q power, then we call x a period of w .Often the word period is used to refer to |x |.
If a word w is a p/q > 1 power, then it begins and ends with somenonempty string. Such a string is also called bordered. Otherwiseit is unbordered.
59 / 201
![Page 60: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/60.jpg)
Unbordered words
The unbordered words play the same role for fractional powers asthe primitive words do for ordinary powers.
60 / 201
![Page 61: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/61.jpg)
Unbordered words
The unbordered words play the same role for fractional powers asthe primitive words do for ordinary powers.
Enumeration of unbordered words is more challenging and there isno simple closed form.
61 / 201
![Page 62: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/62.jpg)
Unbordered words
The unbordered words play the same role for fractional powers asthe primitive words do for ordinary powers.
Enumeration of unbordered words is more challenging and there isno simple closed form.
However, there are asymptotically ckkn such words, where ck is a
constant that tends to 1 as k tends to ∞.
62 / 201
![Page 63: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/63.jpg)
Duval’s conjecture
Let µ(w) be the length of the longest unbordered factor of w .
63 / 201
![Page 64: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/64.jpg)
Duval’s conjecture
Let µ(w) be the length of the longest unbordered factor of w .
Let p(w) be the length of the longest period of w .
64 / 201
![Page 65: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/65.jpg)
Duval’s conjecture
Let µ(w) be the length of the longest unbordered factor of w .
Let p(w) be the length of the longest period of w .
Duval conjectured that if |w | ≥ 3µ(w), then µ(w) = p(w).
65 / 201
![Page 66: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/66.jpg)
Duval’s conjecture
Let µ(w) be the length of the longest unbordered factor of w .
Let p(w) be the length of the longest period of w .
Duval conjectured that if |w | ≥ 3µ(w), then µ(w) = p(w).
This was proved by Harju & Nowotka, and S. Holub. The resulthas been improved to |w | ≥ 3µ(w)− 2 =⇒ µ(w) = p(w).
66 / 201
![Page 67: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/67.jpg)
The Fine-Wilf theorem
Theorem
Let w and x be nonempty words. Let y ∈ w{w , x}ω andz ∈ x{w , x}ω. Then the following conditions are equivalent:
(a) y and z agree on a prefix of length at least|w |+ |x | − gcd(|w |, |x |);
67 / 201
![Page 68: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/68.jpg)
The Fine-Wilf theorem
Theorem
Let w and x be nonempty words. Let y ∈ w{w , x}ω andz ∈ x{w , x}ω. Then the following conditions are equivalent:
(a) y and z agree on a prefix of length at least|w |+ |x | − gcd(|w |, |x |);
(b) wx = xw;
68 / 201
![Page 69: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/69.jpg)
The Fine-Wilf theorem
Theorem
Let w and x be nonempty words. Let y ∈ w{w , x}ω andz ∈ x{w , x}ω. Then the following conditions are equivalent:
(a) y and z agree on a prefix of length at least|w |+ |x | − gcd(|w |, |x |);
(b) wx = xw;
(c) y = z.
(c) =⇒ (a): Trivial.
69 / 201
![Page 70: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/70.jpg)
The Fine-Wilf theorem
Theorem
Let w and x be nonempty words. Let y ∈ w{w , x}ω andz ∈ x{w , x}ω. Then the following conditions are equivalent:
(a) y and z agree on a prefix of length at least|w |+ |x | − gcd(|w |, |x |);
(b) wx = xw;
(c) y = z.
(c) =⇒ (a): Trivial.We’ll prove (a) =⇒ (b) and (b) =⇒ (c).
70 / 201
![Page 71: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/71.jpg)
Fine-Wilf: The proof
Proof.(a) y and z agree on a prefix of length at least|w |+ |x | − gcd(|w |, |x |) =⇒ (b) wx = xw
71 / 201
![Page 72: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/72.jpg)
Fine-Wilf: The proof
Proof.(a) y and z agree on a prefix of length at least|w |+ |x | − gcd(|w |, |x |) =⇒ (b) wx = xw
We prove the contrapositive. Suppose wx 6= xw .
72 / 201
![Page 73: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/73.jpg)
Fine-Wilf: The proof
Proof.(a) y and z agree on a prefix of length at least|w |+ |x | − gcd(|w |, |x |) =⇒ (b) wx = xw
We prove the contrapositive. Suppose wx 6= xw .
Then we prove that y and z differ at a position≤ |w |+ |x | − gcd(|w |, |x |).
73 / 201
![Page 74: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/74.jpg)
Fine-Wilf: The proof
Proof.(a) y and z agree on a prefix of length at least|w |+ |x | − gcd(|w |, |x |) =⇒ (b) wx = xw
We prove the contrapositive. Suppose wx 6= xw .
Then we prove that y and z differ at a position≤ |w |+ |x | − gcd(|w |, |x |).
The proof is by induction on |w |+ |x |.
74 / 201
![Page 75: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/75.jpg)
Fine-Wilf: The proof
Proof.(a) y and z agree on a prefix of length at least|w |+ |x | − gcd(|w |, |x |) =⇒ (b) wx = xw
We prove the contrapositive. Suppose wx 6= xw .
Then we prove that y and z differ at a position≤ |w |+ |x | − gcd(|w |, |x |).
The proof is by induction on |w |+ |x |.
The base case is |w |+ |x | = 2. Then |w | = |x | = 1, and|w |+ |x | − gcd(|w |, |x |) = 1. Since wx 6= xw , we must havew = a, x = b with a 6= b. Then y and z differ at the first position.
75 / 201
![Page 76: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/76.jpg)
Now assume the result is true for |w |+ |x | < k.
76 / 201
![Page 77: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/77.jpg)
Now assume the result is true for |w |+ |x | < k.
We prove it for |w |+ |x | = k.
77 / 201
![Page 78: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/78.jpg)
Now assume the result is true for |w |+ |x | < k.
We prove it for |w |+ |x | = k.
If |w | = |x | then y and z must disagree at the |w |’th position orearlier, for otherwise w = x and wx = xw ; since|w | ≤ |w |+ |x | − gcd(|w |, |x |) = |w |, the result follows.
78 / 201
![Page 79: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/79.jpg)
Now assume the result is true for |w |+ |x | < k.
We prove it for |w |+ |x | = k.
If |w | = |x | then y and z must disagree at the |w |’th position orearlier, for otherwise w = x and wx = xw ; since|w | ≤ |w |+ |x | − gcd(|w |, |x |) = |w |, the result follows.
So, without loss of generality, assume |w | < |x |.
79 / 201
![Page 80: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/80.jpg)
Now assume the result is true for |w |+ |x | < k.
We prove it for |w |+ |x | = k.
If |w | = |x | then y and z must disagree at the |w |’th position orearlier, for otherwise w = x and wx = xw ; since|w | ≤ |w |+ |x | − gcd(|w |, |x |) = |w |, the result follows.
So, without loss of generality, assume |w | < |x |.
If w is not a prefix of x , then y and z disagree on the |w |’thposition or earlier, and again |w | ≤ |w |+ |x | − gcd(|w |, |x |).
80 / 201
![Page 81: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/81.jpg)
Now assume the result is true for |w |+ |x | < k.
We prove it for |w |+ |x | = k.
If |w | = |x | then y and z must disagree at the |w |’th position orearlier, for otherwise w = x and wx = xw ; since|w | ≤ |w |+ |x | − gcd(|w |, |x |) = |w |, the result follows.
So, without loss of generality, assume |w | < |x |.
If w is not a prefix of x , then y and z disagree on the |w |’thposition or earlier, and again |w | ≤ |w |+ |x | − gcd(|w |, |x |).
So w is a proper prefix of x .
81 / 201
![Page 82: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/82.jpg)
Now assume the result is true for |w |+ |x | < k.
We prove it for |w |+ |x | = k.
If |w | = |x | then y and z must disagree at the |w |’th position orearlier, for otherwise w = x and wx = xw ; since|w | ≤ |w |+ |x | − gcd(|w |, |x |) = |w |, the result follows.
So, without loss of generality, assume |w | < |x |.
If w is not a prefix of x , then y and z disagree on the |w |’thposition or earlier, and again |w | ≤ |w |+ |x | − gcd(|w |, |x |).
So w is a proper prefix of x .
Write x = wt for some nonempty word t.
Now any common divisor of |w | and |x | must also divide|x | − |w | = |t|, and similarly any common divisor of both |w | and|t| must also divide |w |+ |t| = |x |. So gcd(|w |, |x |) = gcd(|w |, |t|).
82 / 201
![Page 83: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/83.jpg)
Now wt 6= tw , for otherwise we have wx = wwt = wtw = xw , acontradiction.
83 / 201
![Page 84: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/84.jpg)
Now wt 6= tw , for otherwise we have wx = wwt = wtw = xw , acontradiction.
Then y = ww · · · and z = wt · · · . By induction (since|w |+ |t| < k) w−1y and w−1z disagree at position|w |+ |t| − gcd(|w |, |t|) or earlier.
84 / 201
![Page 85: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/85.jpg)
Now wt 6= tw , for otherwise we have wx = wwt = wtw = xw , acontradiction.
Then y = ww · · · and z = wt · · · . By induction (since|w |+ |t| < k) w−1y and w−1z disagree at position|w |+ |t| − gcd(|w |, |t|) or earlier.
Hence y and z disagree at position2|w |+ |t| − gcd(|w |, |t|) = |w |+ |x | − gcd(|w |, |x |) or earlier.
85 / 201
![Page 86: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/86.jpg)
(b) =⇒ (c): If wx = xw , then by the theorem ofLyndon-Schutzenberger, both w and x are in u+ for some word u.Hence y = uω = z.
86 / 201
![Page 87: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/87.jpg)
Finite Sturmian words
The proof also implies a way to get words that optimally “almostcommute”, in the sense that xw and wx should agree on as long asegment as possible.
87 / 201
![Page 88: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/88.jpg)
Finite Sturmian words
The proof also implies a way to get words that optimally “almostcommute”, in the sense that xw and wx should agree on as long asegment as possible.
Theorem
For each m, n ≥ 1 there exist words x, w of length m, n,respectively, such that xw and wx agree on a prefix of lengthm + n − gcd(m, n)− 1 but differ at position m + n − gcd(m, n).
88 / 201
![Page 89: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/89.jpg)
Finite Sturmian words
The proof also implies a way to get words that optimally “almostcommute”, in the sense that xw and wx should agree on as long asegment as possible.
Theorem
For each m, n ≥ 1 there exist words x, w of length m, n,respectively, such that xw and wx agree on a prefix of lengthm + n − gcd(m, n)− 1 but differ at position m + n − gcd(m, n).
These are the finite Sturmian words.
89 / 201
![Page 90: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/90.jpg)
Finite Sturmian words
The proof also implies a way to get words that optimally “almostcommute”, in the sense that xw and wx should agree on as long asegment as possible.
Theorem
For each m, n ≥ 1 there exist words x, w of length m, n,respectively, such that xw and wx agree on a prefix of lengthm + n − gcd(m, n)− 1 but differ at position m + n − gcd(m, n).
These are the finite Sturmian words.
Many authors have worked on generalizations to multiple periods:Castelli, Mignosi, & Restivo, Simpson & Tijdeman, Constantinescu& Ilie, ...
90 / 201
![Page 91: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/91.jpg)
Patterns and pattern avoidance
The story begins with Axel Thue in 1906.
91 / 201
![Page 92: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/92.jpg)
Patterns and pattern avoidance
The story begins with Axel Thue in 1906.
He noticed that over a 2-letter alphabet, every word of length ≥ 4contains a square: either 02, 12, (01)2 or (10)2.
92 / 201
![Page 93: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/93.jpg)
Patterns and pattern avoidance
The story begins with Axel Thue in 1906.
He noticed that over a 2-letter alphabet, every word of length ≥ 4contains a square: either 02, 12, (01)2 or (10)2.
But over a 3-letter alphabet, it is possible to create arbitrarily longwords (or — what is equivalent — an infinite word) with no squarefactors at all. Such a word is called squarefree.
93 / 201
![Page 94: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/94.jpg)
Patterns and pattern avoidance
The story begins with Axel Thue in 1906.
He noticed that over a 2-letter alphabet, every word of length ≥ 4contains a square: either 02, 12, (01)2 or (10)2.
But over a 3-letter alphabet, it is possible to create arbitrarily longwords (or — what is equivalent — an infinite word) with no squarefactors at all. Such a word is called squarefree.
The easiest way to construct such a sequence was found by Thuein 1912 (and rediscovered many times).
94 / 201
![Page 95: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/95.jpg)
Patterns and pattern avoidance
The story begins with Axel Thue in 1906.
He noticed that over a 2-letter alphabet, every word of length ≥ 4contains a square: either 02, 12, (01)2 or (10)2.
But over a 3-letter alphabet, it is possible to create arbitrarily longwords (or — what is equivalent — an infinite word) with no squarefactors at all. Such a word is called squarefree.
The easiest way to construct such a sequence was found by Thuein 1912 (and rediscovered many times).
It is based on the Thue-Morse sequence.
95 / 201
![Page 96: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/96.jpg)
The Thue-Morse morphism
Morphism: a map h from Σ∗ to ∆∗ such that
h(xy) = h(x)h(y).
96 / 201
![Page 97: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/97.jpg)
The Thue-Morse morphism
Morphism: a map h from Σ∗ to ∆∗ such that
h(xy) = h(x)h(y).
Thue-Morse morphism: µ(0) = 01; µ(1) = 10.
97 / 201
![Page 98: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/98.jpg)
The Thue-Morse morphism
Morphism: a map h from Σ∗ to ∆∗ such that
h(xy) = h(x)h(y).
Thue-Morse morphism: µ(0) = 01; µ(1) = 10.
If Σ = ∆ then we can iterate h.
98 / 201
![Page 99: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/99.jpg)
The Thue-Morse morphism
Morphism: a map h from Σ∗ to ∆∗ such that
h(xy) = h(x)h(y).
Thue-Morse morphism: µ(0) = 01; µ(1) = 10.
If Σ = ∆ then we can iterate h.
We write hi =
i︷ ︸︸ ︷
h(h(h(· · · ))).
99 / 201
![Page 100: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/100.jpg)
Morphic words
If a nonerasing morphism has the property that h(a) = ax , theniterating h produces an infinite word
hω(a) = a x h(x) h2(x) h3(x) · · · .
100 / 201
![Page 101: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/101.jpg)
Morphic words
If a nonerasing morphism has the property that h(a) = ax , theniterating h produces an infinite word
hω(a) = a x h(x) h2(x) h3(x) · · · .
If we do this with µ we get the Thue-Morse word:
101 / 201
![Page 102: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/102.jpg)
Morphic words
If a nonerasing morphism has the property that h(a) = ax , theniterating h produces an infinite word
hω(a) = a x h(x) h2(x) h3(x) · · · .
If we do this with µ we get the Thue-Morse word:
t = µω(0) = 0110100110010110 · · · .
102 / 201
![Page 103: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/103.jpg)
Morphic words
If a nonerasing morphism has the property that h(a) = ax , theniterating h produces an infinite word
hω(a) = a x h(x) h2(x) h3(x) · · · .
If we do this with µ we get the Thue-Morse word:
t = µω(0) = 0110100110010110 · · · .
Also rediscovered by Marston Morse, Max Euwe, Solomon Arshon,and the Danish composer Per Nørgard.
103 / 201
![Page 104: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/104.jpg)
Properties of the Thue-Morse word
An overlap is a word of the form axaxa, where a is a single letterand x is a word.
104 / 201
![Page 105: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/105.jpg)
Properties of the Thue-Morse word
An overlap is a word of the form axaxa, where a is a single letterand x is a word.
An example in English is alfalfa (take x = lf).
105 / 201
![Page 106: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/106.jpg)
Properties of the Thue-Morse word
An overlap is a word of the form axaxa, where a is a single letterand x is a word.
An example in English is alfalfa (take x = lf).
Thus an overlap is just slightly more than a square.
106 / 201
![Page 107: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/107.jpg)
Properties of the Thue-Morse word
An overlap is a word of the form axaxa, where a is a single letterand x is a word.
An example in English is alfalfa (take x = lf).
Thus an overlap is just slightly more than a square.
It is also called a 2+ power.
107 / 201
![Page 108: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/108.jpg)
Properties of the Thue-Morse word
An overlap is a word of the form axaxa, where a is a single letterand x is a word.
An example in English is alfalfa (take x = lf).
Thus an overlap is just slightly more than a square.
It is also called a 2+ power.
Theorem
The Thue-Morse word t is overlap-free.
108 / 201
![Page 109: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/109.jpg)
From overlap-free to squarefree
We can construct a squarefree word from t, as follows:
109 / 201
![Page 110: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/110.jpg)
From overlap-free to squarefree
We can construct a squarefree word from t, as follows:
Count the number of 1’s in t between consecutive 0’s:
110 / 201
![Page 111: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/111.jpg)
From overlap-free to squarefree
We can construct a squarefree word from t, as follows:
Count the number of 1’s in t between consecutive 0’s:
We get:
111 / 201
![Page 112: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/112.jpg)
From overlap-free to squarefree
We can construct a squarefree word from t, as follows:
Count the number of 1’s in t between consecutive 0’s:
We get:
0
2︷︸︸︷
11 0
1︷︸︸︷
1 0
0︷︸︸︷
0
2︷︸︸︷
11 0
0︷︸︸︷
0
1︷︸︸︷
1 0
2︷︸︸︷
11 0
1︷︸︸︷
1 0
0︷︸︸︷
0
1︷︸︸︷
1 0
2︷︸︸︷
11 · · ·
112 / 201
![Page 113: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/113.jpg)
From overlap-free to squarefree
We can construct a squarefree word from t, as follows:
Count the number of 1’s in t between consecutive 0’s:
We get:
0
2︷︸︸︷
11 0
1︷︸︸︷
1 0
0︷︸︸︷
0
2︷︸︸︷
11 0
0︷︸︸︷
0
1︷︸︸︷
1 0
2︷︸︸︷
11 0
1︷︸︸︷
1 0
0︷︸︸︷
0
1︷︸︸︷
1 0
2︷︸︸︷
11 · · ·
This is squarefree, as a square in this word implies and overlap inthe Thue-Morse word.
113 / 201
![Page 114: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/114.jpg)
Enumeration of power-free words
How many squarefree words are there?
114 / 201
![Page 115: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/115.jpg)
Enumeration of power-free words
How many squarefree words are there?
Infinite - countable or uncountable
115 / 201
![Page 116: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/116.jpg)
Enumeration of power-free words
How many squarefree words are there?
Infinite - countable or uncountable
Finite - polynomially-many or exponentially-many of length n?
116 / 201
![Page 117: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/117.jpg)
Enumeration of power-free words
How many squarefree words are there?
Infinite - countable or uncountable
Finite - polynomially-many or exponentially-many of length n?
Same question can be asked for the overlap-free words.
117 / 201
![Page 118: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/118.jpg)
Enumeration of power-free words
How many squarefree words are there?
Infinite - countable or uncountable
Finite - polynomially-many or exponentially-many of length n?
Same question can be asked for the overlap-free words.
For overlap-free words over {0, 1} there is a factorization theoremof Restivo and Salemi that implies only polynomially-many oflength n.
118 / 201
![Page 119: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/119.jpg)
Enumeration of power-free words
How many squarefree words are there?
Infinite - countable or uncountable
Finite - polynomially-many or exponentially-many of length n?
Same question can be asked for the overlap-free words.
For overlap-free words over {0, 1} there is a factorization theoremof Restivo and Salemi that implies only polynomially-many oflength n.
For squarefree words over {0, 1, 2} there are exponentially many.
119 / 201
![Page 120: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/120.jpg)
Dejean’s Conjecture
Given an alphabet Σ of cardinality k, we can try to find the optimal(fractional) exponent αk avoidable by infinite words over Σ.
120 / 201
![Page 121: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/121.jpg)
Dejean’s Conjecture
Given an alphabet Σ of cardinality k, we can try to find the optimal(fractional) exponent αk avoidable by infinite words over Σ.
For k = 2 we have already seen that overlaps are avoidable andsquares are not. So α2 = 2.
121 / 201
![Page 122: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/122.jpg)
Dejean’s Conjecture
Given an alphabet Σ of cardinality k, we can try to find the optimal(fractional) exponent αk avoidable by infinite words over Σ.
For k = 2 we have already seen that overlaps are avoidable andsquares are not. So α2 = 2.
Dejean (1972) showed that α3 = 7/4 and conjectured thatα4 = 7/5 and αk = k/(k − 1) for k ≥ 5.
122 / 201
![Page 123: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/123.jpg)
Dejean’s Conjecture
Given an alphabet Σ of cardinality k, we can try to find the optimal(fractional) exponent αk avoidable by infinite words over Σ.
For k = 2 we have already seen that overlaps are avoidable andsquares are not. So α2 = 2.
Dejean (1972) showed that α3 = 7/4 and conjectured thatα4 = 7/5 and αk = k/(k − 1) for k ≥ 5.
This conjecture has been proven by the combined efforts ofPansiot, Moulin-Ollagnier, Currie & Mohammad-Noori, Carpi,Currie & Rampersad, and Rao.
123 / 201
![Page 124: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/124.jpg)
Dejean’s Conjecture
Given an alphabet Σ of cardinality k, we can try to find the optimal(fractional) exponent αk avoidable by infinite words over Σ.
For k = 2 we have already seen that overlaps are avoidable andsquares are not. So α2 = 2.
Dejean (1972) showed that α3 = 7/4 and conjectured thatα4 = 7/5 and αk = k/(k − 1) for k ≥ 5.
This conjecture has been proven by the combined efforts ofPansiot, Moulin-Ollagnier, Currie & Mohammad-Noori, Carpi,Currie & Rampersad, and Rao.
Still open: many other variants of Dejean where the length of theperiod is also taken into account.
124 / 201
![Page 125: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/125.jpg)
More general patterns
Instead of avoiding xx or axaxa, we can try to avoid more generalpatterns.
125 / 201
![Page 126: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/126.jpg)
More general patterns
Instead of avoiding xx or axaxa, we can try to avoid more generalpatterns.
“Avoiding the pattern α” means constructing an infinite word x
such that, for all non-erasing morphisms h, the word h(α) is not afactor of x.
126 / 201
![Page 127: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/127.jpg)
More general patterns
Instead of avoiding xx or axaxa, we can try to avoid more generalpatterns.
“Avoiding the pattern α” means constructing an infinite word x
such that, for all non-erasing morphisms h, the word h(α) is not afactor of x.
Not all patterns are avoidable — even if the alphabet is arbitrarilylarge.
127 / 201
![Page 128: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/128.jpg)
More general patterns
Instead of avoiding xx or axaxa, we can try to avoid more generalpatterns.
“Avoiding the pattern α” means constructing an infinite word x
such that, for all non-erasing morphisms h, the word h(α) is not afactor of x.
Not all patterns are avoidable — even if the alphabet is arbitrarilylarge.
For example - it is impossible to avoid xyx , since every sufficientlylong string z will contain three occurrences of some letter a, sayz = rasatau, and then we can let x = a, y = sat, both x and y arenonempty.
128 / 201
![Page 129: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/129.jpg)
Avoiding general patterns
Given a pattern, it is decidable (via Zimin’s algorithm) if it isavoidable over some alphabet.
129 / 201
![Page 130: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/130.jpg)
Avoiding general patterns
Given a pattern, it is decidable (via Zimin’s algorithm) if it isavoidable over some alphabet.
However, we do not have a general procedure to decide if a givenpattern is avoidable over a fixed alphabet.
130 / 201
![Page 131: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/131.jpg)
Abelian powers
Instead of avoiding xx , we can consider trying to avoid other kindsof patterns: the so-called abelian powers.
131 / 201
![Page 132: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/132.jpg)
Abelian powers
Instead of avoiding xx , we can consider trying to avoid other kindsof patterns: the so-called abelian powers.
An abelian square is a nonempty word of the form xx ′, where x ′ isa permutation of x .
132 / 201
![Page 133: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/133.jpg)
Abelian powers
Instead of avoiding xx , we can consider trying to avoid other kindsof patterns: the so-called abelian powers.
An abelian square is a nonempty word of the form xx ′, where x ′ isa permutation of x .
For example, interessierten is an abelian square in German, assierten is a permutation of interes.
133 / 201
![Page 134: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/134.jpg)
Abelian powers
Instead of avoiding xx , we can consider trying to avoid other kindsof patterns: the so-called abelian powers.
An abelian square is a nonempty word of the form xx ′, where x ′ isa permutation of x .
For example, interessierten is an abelian square in German, assierten is a permutation of interes.
In a similar way, we can define abelian cubes as words of the formxx ′x ′′ where both x ′ and x ′′ are permutations of x .
134 / 201
![Page 135: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/135.jpg)
Abelian powers: summary of results
it is possible to avoid abelian squares over a 4-letter alphabet, andthis is optimal;
135 / 201
![Page 136: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/136.jpg)
Abelian powers: summary of results
it is possible to avoid abelian squares over a 4-letter alphabet, andthis is optimal;
it is possible to avoid abelian cubes over a 3-letter alphabet, andthis is optimal;
136 / 201
![Page 137: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/137.jpg)
Abelian powers: summary of results
it is possible to avoid abelian squares over a 4-letter alphabet, andthis is optimal;
it is possible to avoid abelian cubes over a 3-letter alphabet, andthis is optimal;
it is possible to avoid abelian fourth powers over a 2-letteralphabet, and this is optimal.
137 / 201
![Page 138: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/138.jpg)
Abelian powers: summary of results
it is possible to avoid abelian squares over a 4-letter alphabet, andthis is optimal;
it is possible to avoid abelian cubes over a 3-letter alphabet, andthis is optimal;
it is possible to avoid abelian fourth powers over a 2-letteralphabet, and this is optimal.
An open problem: is it possible to avoid, over a finite subset of N,patterns of the form xx ′ where |x | = |x ′| and
∑x =
∑x ′?
138 / 201
![Page 139: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/139.jpg)
Abelian powers: summary of results
it is possible to avoid abelian squares over a 4-letter alphabet, andthis is optimal;
it is possible to avoid abelian cubes over a 3-letter alphabet, andthis is optimal;
it is possible to avoid abelian fourth powers over a 2-letteralphabet, and this is optimal.
An open problem: is it possible to avoid, over a finite subset of N,patterns of the form xx ′ where |x | = |x ′| and
∑x =
∑x ′?
Also still open: fractional version of abelian powers
139 / 201
![Page 140: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/140.jpg)
A generalization of abelian powers
involution: h2(x) = x for all words x
140 / 201
![Page 141: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/141.jpg)
A generalization of abelian powers
involution: h2(x) = x for all words x
involutions can be morphic (h(xy) = h(x)h(y)) or antimorphic(h(xy) = h(y)h(x))
141 / 201
![Page 142: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/142.jpg)
A generalization of abelian powers
involution: h2(x) = x for all words x
involutions can be morphic (h(xy) = h(x)h(y)) or antimorphic(h(xy) = h(y)h(x))
One antimorphism of biological interest: reverse string and applymap A↔ T , G ↔ C .
142 / 201
![Page 143: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/143.jpg)
A generalization of abelian powers
involution: h2(x) = x for all words x
involutions can be morphic (h(xy) = h(x)h(y)) or antimorphic(h(xy) = h(y)h(x))
One antimorphism of biological interest: reverse string and applymap A↔ T , G ↔ C .
Can avoid some patterns involving involution, but not others
143 / 201
![Page 144: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/144.jpg)
Equations in words
Example 1:
abX = Xba.
144 / 201
![Page 145: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/145.jpg)
Equations in words
Example 1:
abX = Xba.
The only solutions are X ∈ (ab)∗a.
145 / 201
![Page 146: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/146.jpg)
Equations in words
Example 1:
abX = Xba.
The only solutions are X ∈ (ab)∗a.
Example 2:
XaXbY = aXYbX
146 / 201
![Page 147: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/147.jpg)
Equations in words
Example 1:
abX = Xba.
The only solutions are X ∈ (ab)∗a.
Example 2:
XaXbY = aXYbX
The solutions are X = ai , Y = (aib)jai for i ≥ 0, j ≥ 0.
147 / 201
![Page 148: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/148.jpg)
Equations in words
Example 3: Fermat’s equation for words: x iy j = zk
148 / 201
![Page 149: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/149.jpg)
Equations in words
Example 3: Fermat’s equation for words: x iy j = zk
The only solutions for i , j , k ≥ 2 are when x , y , z are all powers ofa third word.
149 / 201
![Page 150: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/150.jpg)
Equations in words
Example 3: Fermat’s equation for words: x iy j = zk
The only solutions for i , j , k ≥ 2 are when x , y , z are all powers ofa third word.
Example 4:
XYZ = ZVX : many solutions, but not expressible by formula withinteger parameters.
150 / 201
![Page 151: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/151.jpg)
Equations in words
More generally, given an equation in words and constants, there isan algorithm (Makanin’s algorithm) that is guaranteed to find asolution if one exists.
151 / 201
![Page 152: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/152.jpg)
Equations in words
More generally, given an equation in words and constants, there isan algorithm (Makanin’s algorithm) that is guaranteed to find asolution if one exists.
Satisfiability in PSPACE: Plandowski (2004)
152 / 201
![Page 153: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/153.jpg)
Equations in words
More generally, given an equation in words and constants, there isan algorithm (Makanin’s algorithm) that is guaranteed to find asolution if one exists.
Satisfiability in PSPACE: Plandowski (2004)
Finiteness of solutions: Plandowski (2006) but complete proofs notyet published.
153 / 201
![Page 154: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/154.jpg)
Kinds of infinite words
pure morphic: obtained by iterating a morphism
154 / 201
![Page 155: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/155.jpg)
Kinds of infinite words
pure morphic: obtained by iterating a morphism
Examples:
155 / 201
![Page 156: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/156.jpg)
Kinds of infinite words
pure morphic: obtained by iterating a morphism
Examples:
the Thue-Morse word, obtained by iterating 0→ 01, 1→ 10
156 / 201
![Page 157: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/157.jpg)
Kinds of infinite words
pure morphic: obtained by iterating a morphism
Examples:
the Thue-Morse word, obtained by iterating 0→ 01, 1→ 10
the Fibonacci word, obtained by iterating 0→ 01, 1→ 0
157 / 201
![Page 158: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/158.jpg)
Kinds of infinite words
pure morphic: obtained by iterating a morphism
Examples:
the Thue-Morse word, obtained by iterating 0→ 01, 1→ 10
the Fibonacci word, obtained by iterating 0→ 01, 1→ 0
morphic: obtained by iterating a morphism, then applying a coding(letter-to-letter morphism)
158 / 201
![Page 159: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/159.jpg)
Kinds of infinite words
pure morphic: obtained by iterating a morphism
Examples:
the Thue-Morse word, obtained by iterating 0→ 01, 1→ 10
the Fibonacci word, obtained by iterating 0→ 01, 1→ 0
morphic: obtained by iterating a morphism, then applying a coding(letter-to-letter morphism)
fixed point of uniform morphism: like the Thue-Morse word
159 / 201
![Page 160: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/160.jpg)
Kinds of infinite words
pure morphic: obtained by iterating a morphism
Examples:
the Thue-Morse word, obtained by iterating 0→ 01, 1→ 10
the Fibonacci word, obtained by iterating 0→ 01, 1→ 0
morphic: obtained by iterating a morphism, then applying a coding(letter-to-letter morphism)
fixed point of uniform morphism: like the Thue-Morse word
automatic: image, under a coding, of a fixed point of a uniformmorphism
160 / 201
![Page 161: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/161.jpg)
Other important infinite words
Sturmian words: exactly n + 1 factors of length n
161 / 201
![Page 162: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/162.jpg)
Other important infinite words
Sturmian words: exactly n + 1 factors of length nMany other characterizations:
162 / 201
![Page 163: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/163.jpg)
Other important infinite words
Sturmian words: exactly n + 1 factors of length nMany other characterizations:
balanced: any two words w , x of length n have δ(x , y) = 1,where δ(x , y) = ||x |1 − |y |1|
163 / 201
![Page 164: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/164.jpg)
Other important infinite words
Sturmian words: exactly n + 1 factors of length nMany other characterizations:
balanced: any two words w , x of length n have δ(x , y) = 1,where δ(x , y) = ||x |1 − |y |1|
of the form (bα(n + 1) + γc − bαn + γc)n≥1 for real α, γ
164 / 201
![Page 165: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/165.jpg)
Other important infinite words
Sturmian words: exactly n + 1 factors of length nMany other characterizations:
balanced: any two words w , x of length n have δ(x , y) = 1,where δ(x , y) = ||x |1 − |y |1|
of the form (bα(n + 1) + γc − bαn + γc)n≥1 for real α, γ
Episturmian words: natural generalization of Sturmian words tolarger alphabets
165 / 201
![Page 166: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/166.jpg)
More infinite words
Toeplitz words: generated by starting with a periodic word with“holes”; then inserting another periodic word with holes into that,etc.
166 / 201
![Page 167: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/167.jpg)
More infinite words
Toeplitz words: generated by starting with a periodic word with“holes”; then inserting another periodic word with holes into that,etc.
Paperfolding words: generated by iterated folding of a piece ofpaper
167 / 201
![Page 168: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/168.jpg)
More infinite words
Toeplitz words: generated by starting with a periodic word with“holes”; then inserting another periodic word with holes into that,etc.
Paperfolding words: generated by iterated folding of a piece ofpaper
Xn+1 = XnaXnR
168 / 201
![Page 169: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/169.jpg)
More infinite words
Toeplitz words: generated by starting with a periodic word with“holes”; then inserting another periodic word with holes into that,etc.
Paperfolding words: generated by iterated folding of a piece ofpaper
Xn+1 = XnaXnR
Kolakoski’s word: generated by applying a transducer iteratively
169 / 201
![Page 170: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/170.jpg)
More infinite words
Toeplitz words: generated by starting with a periodic word with“holes”; then inserting another periodic word with holes into that,etc.
Paperfolding words: generated by iterated folding of a piece ofpaper
Xn+1 = XnaXnR
Kolakoski’s word: generated by applying a transducer iteratively
The sequence 1221121221 · · · that encodes its own sequenceof run lengths
170 / 201
![Page 171: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/171.jpg)
Properties of infinite words
recurrence - every factor that occurs, occurs infinitely often
171 / 201
![Page 172: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/172.jpg)
Properties of infinite words
recurrence - every factor that occurs, occurs infinitely often
uniform recurrence - recurrent, plus distance between twoconsecutive occurrences of the same factor of length n is bounded,for all n
172 / 201
![Page 173: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/173.jpg)
Subword complexity
“subword” complexity - given an infinite word w, count thenumber of distinct factors of length n in w
173 / 201
![Page 174: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/174.jpg)
Subword complexity
“subword” complexity - given an infinite word w, count thenumber of distinct factors of length n in w
O(n) for automatic sequences
174 / 201
![Page 175: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/175.jpg)
Subword complexity
“subword” complexity - given an infinite word w, count thenumber of distinct factors of length n in w
O(n) for automatic sequences
n + 1 for Sturmian words
175 / 201
![Page 176: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/176.jpg)
Subword complexity
“subword” complexity - given an infinite word w, count thenumber of distinct factors of length n in w
O(n) for automatic sequences
n + 1 for Sturmian words
O(n2) for morphic words
176 / 201
![Page 177: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/177.jpg)
Subword complexity
“subword” complexity - given an infinite word w, count thenumber of distinct factors of length n in w
O(n) for automatic sequences
n + 1 for Sturmian words
O(n2) for morphic words
A classification of possible growth rates exists
177 / 201
![Page 178: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/178.jpg)
Automatic sequences
A deterministic finite automaton with output (DFAO) is a6-tuple: (Q,Σ, δ, q0,∆, τ), where ∆ is the finite outputalphabet and τ : Q → ∆ is the output mapping.
178 / 201
![Page 179: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/179.jpg)
Automatic sequences
A deterministic finite automaton with output (DFAO) is a6-tuple: (Q,Σ, δ, q0,∆, τ), where ∆ is the finite outputalphabet and τ : Q → ∆ is the output mapping.Next, we decide on a integer base k ≥ 2 and represent n as astring of symbols over the alphabet Σ = {0, 1, 2, . . . , k − 1}.
179 / 201
![Page 180: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/180.jpg)
Automatic sequences
A deterministic finite automaton with output (DFAO) is a6-tuple: (Q,Σ, δ, q0,∆, τ), where ∆ is the finite outputalphabet and τ : Q → ∆ is the output mapping.Next, we decide on a integer base k ≥ 2 and represent n as astring of symbols over the alphabet Σ = {0, 1, 2, . . . , k − 1}.To compute fn, given an automaton M, express n in base-k,say,
arar−1 · · · a1a0,
and compute
fn = τ(δ(q0, arar−1 · · · a1a0)).
180 / 201
![Page 181: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/181.jpg)
Automatic sequences
A deterministic finite automaton with output (DFAO) is a6-tuple: (Q,Σ, δ, q0,∆, τ), where ∆ is the finite outputalphabet and τ : Q → ∆ is the output mapping.Next, we decide on a integer base k ≥ 2 and represent n as astring of symbols over the alphabet Σ = {0, 1, 2, . . . , k − 1}.To compute fn, given an automaton M, express n in base-k,say,
arar−1 · · · a1a0,
and compute
fn = τ(δ(q0, arar−1 · · · a1a0)).
Any sequence that can be computed in this way is said to bek-automatic.
181 / 201
![Page 182: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/182.jpg)
The Thue-Morse automaton
The word t is computed by the following DFAO:
182 / 201
![Page 183: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/183.jpg)
The Thue-Morse automaton
The word t is computed by the following DFAO:
0 01
1
0 1
183 / 201
![Page 184: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/184.jpg)
Robustness
the order in which the base-k digits are fed into theautomaton in does not matter (provided it is fixed for all n);
184 / 201
![Page 185: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/185.jpg)
Robustness
the order in which the base-k digits are fed into theautomaton in does not matter (provided it is fixed for all n);
other representations also work (such as expansion inbase-(−k));
185 / 201
![Page 186: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/186.jpg)
Robustness
the order in which the base-k digits are fed into theautomaton in does not matter (provided it is fixed for all n);
other representations also work (such as expansion inbase-(−k));
automatic sequences are closed under many operations, suchas shift, periodic deletion, q-block compression, and q-blocksubstitution.
186 / 201
![Page 187: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/187.jpg)
Robustness
the order in which the base-k digits are fed into theautomaton in does not matter (provided it is fixed for all n);
other representations also work (such as expansion inbase-(−k));
automatic sequences are closed under many operations, suchas shift, periodic deletion, q-block compression, and q-blocksubstitution.
if a symbol in an automatic sequence occurs with well-definedfrequency r , then r is rational.
187 / 201
![Page 188: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/188.jpg)
Christol’s theorem
Theorem
(Christol [1980]). Let (un)n≥0 be a sequence over
Σ = {0, 1, . . . , p − 1},
where p is a prime. Then the formal power seriesU(X ) =
∑
n≥0 unXn is algebraic over GF (p)[X ] if and only if
(un)n≥0 is p-automatic.
188 / 201
![Page 189: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/189.jpg)
Christol’s theorem: example
Let (tn)n≥0 denote the Thue-Morse sequence.
189 / 201
![Page 190: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/190.jpg)
Christol’s theorem: example
Let (tn)n≥0 denote the Thue-Morse sequence.Then tn = sum of the bits in the binary expansion of n, mod 2.
190 / 201
![Page 191: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/191.jpg)
Christol’s theorem: example
Let (tn)n≥0 denote the Thue-Morse sequence.Then tn = sum of the bits in the binary expansion of n, mod 2.Also t2n ≡ tn and t2n+1 ≡ tn + 1. If we set A(X ) =
∑
n≥0 tnXn,
then
191 / 201
![Page 192: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/192.jpg)
Christol’s theorem: example
Let (tn)n≥0 denote the Thue-Morse sequence.Then tn = sum of the bits in the binary expansion of n, mod 2.Also t2n ≡ tn and t2n+1 ≡ tn + 1. If we set A(X ) =
∑
n≥0 tnXn,
then
A(X ) =∑
n≥0
t2nX2n +
∑
n≥0
t2n+1X2n+1
192 / 201
![Page 193: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/193.jpg)
Christol’s theorem: example
Let (tn)n≥0 denote the Thue-Morse sequence.Then tn = sum of the bits in the binary expansion of n, mod 2.Also t2n ≡ tn and t2n+1 ≡ tn + 1. If we set A(X ) =
∑
n≥0 tnXn,
then
A(X ) =∑
n≥0
t2nX2n +
∑
n≥0
t2n+1X2n+1
=∑
n≥0
tnX2n + X
∑
n≥0
tnX2n + X
∑
n≥0
X 2n
193 / 201
![Page 194: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/194.jpg)
Christol’s theorem: example
Let (tn)n≥0 denote the Thue-Morse sequence.Then tn = sum of the bits in the binary expansion of n, mod 2.Also t2n ≡ tn and t2n+1 ≡ tn + 1. If we set A(X ) =
∑
n≥0 tnXn,
then
A(X ) =∑
n≥0
t2nX2n +
∑
n≥0
t2n+1X2n+1
=∑
n≥0
tnX2n + X
∑
n≥0
tnX2n + X
∑
n≥0
X 2n
= A(X 2) + XA(X 2) + X/(1− X 2)
194 / 201
![Page 195: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/195.jpg)
Christol’s theorem: example
Let (tn)n≥0 denote the Thue-Morse sequence.Then tn = sum of the bits in the binary expansion of n, mod 2.Also t2n ≡ tn and t2n+1 ≡ tn + 1. If we set A(X ) =
∑
n≥0 tnXn,
then
A(X ) =∑
n≥0
t2nX2n +
∑
n≥0
t2n+1X2n+1
=∑
n≥0
tnX2n + X
∑
n≥0
tnX2n + X
∑
n≥0
X 2n
= A(X 2) + XA(X 2) + X/(1− X 2)
= A(X )2(1 + X ) + X/(1 + X )2.
195 / 201
![Page 196: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/196.jpg)
Christol’s theorem: example
Let (tn)n≥0 denote the Thue-Morse sequence.Then tn = sum of the bits in the binary expansion of n, mod 2.Also t2n ≡ tn and t2n+1 ≡ tn + 1. If we set A(X ) =
∑
n≥0 tnXn,
then
A(X ) =∑
n≥0
t2nX2n +
∑
n≥0
t2n+1X2n+1
=∑
n≥0
tnX2n + X
∑
n≥0
tnX2n + X
∑
n≥0
X 2n
= A(X 2) + XA(X 2) + X/(1− X 2)
= A(X )2(1 + X ) + X/(1 + X )2.
Hence (1 + X )3A2 + (1 + X )2A+ X = 0.196 / 201
![Page 197: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/197.jpg)
Open Problems
Is the set of primitive words over {0, 1} context-free? (Almostcertainly not.)
197 / 201
![Page 198: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/198.jpg)
Open Problems
Is the set of primitive words over {0, 1} context-free? (Almostcertainly not.)
What are the frequencies of letters in Kolakoski’s word? Dothey exist? Are they equal to 1/2?
198 / 201
![Page 199: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/199.jpg)
Open Problems
Is the set of primitive words over {0, 1} context-free? (Almostcertainly not.)
What are the frequencies of letters in Kolakoski’s word? Dothey exist? Are they equal to 1/2?
Nivat’s conjecture: extension of periodicity to 2-dimensionalarrays
199 / 201
![Page 200: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/200.jpg)
Open Problems
Is the set of primitive words over {0, 1} context-free? (Almostcertainly not.)
What are the frequencies of letters in Kolakoski’s word? Dothey exist? Are they equal to 1/2?
Nivat’s conjecture: extension of periodicity to 2-dimensionalarrays
Is there a word over a finite subset of N that avoids xx ′ with|x | = |x ′| and
∑x =
∑x ′?
200 / 201
![Page 201: Combinatorics on Words: An Introductionshallit/Talks/introcw.pdf · Nowassumetheresultistrueforjwj+jxj](https://reader035.vdocument.in/reader035/viewer/2022062602/5edc88b3ad6a402d66673bbe/html5/thumbnails/201.jpg)
For Further Reading
M. Lothaire, Combinatorics on Words, Cambridge, 1997(reprint)
M. Lothaire, Algebraic Combinatorics on Words, Cambridge,2002
M. Lothaire, Applied Combinatorics on Words, Cambridge,2005
V. Berthe, M. Rigo, Combinatorics, Automata, and NumberTheory, Cambridge, 2010
J.-P. Allouche and J. Shallit, Automatic Sequences: Theory,Applications, Generalizations, Cambridge, 2003.
201 / 201