A glimpse of combinatorics on words

Transcription

1 A glimpse of combinatorics on words Benny George K Tata Institute of Fundamental Research,Mumbai India Indo Russian Workshop 2008 Ural State Univeristy Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 1 / 13

2 Outline Introduction 1 Introduction 2 Fine and Wilf s Theorem Pattern Avoidance Theorems 3 Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 2 / 13

3 Outline Introduction 1 Introduction 2 Fine and Wilf s Theorem Pattern Avoidance Theorems 3 Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 2 / 13

4 Outline Introduction 1 Introduction 2 Fine and Wilf s Theorem Pattern Avoidance Theorems 3 Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 2 / 13

5 Outline Introduction 1 Introduction 2 Fine and Wilf s Theorem Pattern Avoidance Theorems 3 Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 3 / 13

6 Basic Terminology A is a finite alphabet. A + (resp. A )is the free semigroup (resp. monoid) generated by A Elements of A are called words. Subsets of A are called languages. Lower cases letters x, y, z,... denotes words and upper case letters X, Y, Z... denotes languages. Lower case letters a, b, c,... are used for constants. 1 denotes the empty word. w is length of word w. For two words u and v, u is a prefix(suffix, factor) of v if there exists x, y such that v = ux(v = xu, v = xuy). Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 4 / 13

7 Basic Terminology A is a finite alphabet. A + (resp. A )is the free semigroup (resp. monoid) generated by A Elements of A are called words. Subsets of A are called languages. Lower cases letters x, y, z,... denotes words and upper case letters X, Y, Z... denotes languages. Lower case letters a, b, c,... are used for constants. 1 denotes the empty word. w is length of word w. For two words u and v, u is a prefix(suffix, factor) of v if there exists x, y such that v = ux(v = xu, v = xuy). Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 4 / 13

8 Basic Terminology A is a finite alphabet. A + (resp. A )is the free semigroup (resp. monoid) generated by A Elements of A are called words. Subsets of A are called languages. Lower cases letters x, y, z,... denotes words and upper case letters X, Y, Z... denotes languages. Lower case letters a, b, c,... are used for constants. 1 denotes the empty word. w is length of word w. For two words u and v, u is a prefix(suffix, factor) of v if there exists x, y such that v = ux(v = xu, v = xuy). Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 4 / 13

9 Basic Terminology A is a finite alphabet. A + (resp. A )is the free semigroup (resp. monoid) generated by A Elements of A are called words. Subsets of A are called languages. Lower cases letters x, y, z,... denotes words and upper case letters X, Y, Z... denotes languages. Lower case letters a, b, c,... are used for constants. 1 denotes the empty word. w is length of word w. For two words u and v, u is a prefix(suffix, factor) of v if there exists x, y such that v = ux(v = xu, v = xuy). Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 4 / 13

10 Basic Terminology A is a finite alphabet. A + (resp. A )is the free semigroup (resp. monoid) generated by A Elements of A are called words. Subsets of A are called languages. Lower cases letters x, y, z,... denotes words and upper case letters X, Y, Z... denotes languages. Lower case letters a, b, c,... are used for constants. 1 denotes the empty word. w is length of word w. For two words u and v, u is a prefix(suffix, factor) of v if there exists x, y such that v = ux(v = xu, v = xuy). Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 4 / 13

11 Basic Terminology A is a finite alphabet. A + (resp. A )is the free semigroup (resp. monoid) generated by A Elements of A are called words. Subsets of A are called languages. Lower cases letters x, y, z,... denotes words and upper case letters X, Y, Z... denotes languages. Lower case letters a, b, c,... are used for constants. 1 denotes the empty word. w is length of word w. For two words u and v, u is a prefix(suffix, factor) of v if there exists x, y such that v = ux(v = xu, v = xuy). Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 4 / 13

12 Basic Terminology h is a morphism from A to B if h(uv) = h(u)h(v) φ(a) = ab, φ(b) = a This is called fibonacci morphism. Let f = φ (a) f 0 = a,f 1 = ab and f n+1 = f n f n 1. If w = xu k y, we say w has a repetition of order k If a words doesn t contain a repetition of order k its is called k free. A word is called primitive if it is not of the form w k for k > 1. An integer p is a period of w = a 1... a n if a i = a p+i. The smallest period is often called the period. Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 5 / 13

13 Basic Terminology h is a morphism from A to B if h(uv) = h(u)h(v) φ(a) = ab, φ(b) = a This is called fibonacci morphism. Let f = φ (a) f 0 = a,f 1 = ab and f n+1 = f n f n 1. If w = xu k y, we say w has a repetition of order k If a words doesn t contain a repetition of order k its is called k free. A word is called primitive if it is not of the form w k for k > 1. An integer p is a period of w = a 1... a n if a i = a p+i. The smallest period is often called the period. Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 5 / 13

14 Basic Terminology h is a morphism from A to B if h(uv) = h(u)h(v) φ(a) = ab, φ(b) = a This is called fibonacci morphism. Let f = φ (a) f 0 = a,f 1 = ab and f n+1 = f n f n 1. If w = xu k y, we say w has a repetition of order k If a words doesn t contain a repetition of order k its is called k free. A word is called primitive if it is not of the form w k for k > 1. An integer p is a period of w = a 1... a n if a i = a p+i. The smallest period is often called the period. Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 5 / 13

15 Basic Terminology h is a morphism from A to B if h(uv) = h(u)h(v) φ(a) = ab, φ(b) = a This is called fibonacci morphism. Let f = φ (a) f 0 = a,f 1 = ab and f n+1 = f n f n 1. If w = xu k y, we say w has a repetition of order k If a words doesn t contain a repetition of order k its is called k free. A word is called primitive if it is not of the form w k for k > 1. An integer p is a period of w = a 1... a n if a i = a p+i. The smallest period is often called the period. Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 5 / 13

16 Basic Terminology h is a morphism from A to B if h(uv) = h(u)h(v) φ(a) = ab, φ(b) = a This is called fibonacci morphism. Let f = φ (a) f 0 = a,f 1 = ab and f n+1 = f n f n 1. If w = xu k y, we say w has a repetition of order k If a words doesn t contain a repetition of order k its is called k free. A word is called primitive if it is not of the form w k for k > 1. An integer p is a period of w = a 1... a n if a i = a p+i. The smallest period is often called the period. Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 5 / 13

17 Basic Terminology h is a morphism from A to B if h(uv) = h(u)h(v) φ(a) = ab, φ(b) = a This is called fibonacci morphism. Let f = φ (a) f 0 = a,f 1 = ab and f n+1 = f n f n 1. If w = xu k y, we say w has a repetition of order k If a words doesn t contain a repetition of order k its is called k free. A word is called primitive if it is not of the form w k for k > 1. An integer p is a period of w = a 1... a n if a i = a p+i. The smallest period is often called the period. Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 5 / 13

18 Basic Terminology h is a morphism from A to B if h(uv) = h(u)h(v) φ(a) = ab, φ(b) = a This is called fibonacci morphism. Let f = φ (a) f 0 = a,f 1 = ab and f n+1 = f n f n 1. If w = xu k y, we say w has a repetition of order k If a words doesn t contain a repetition of order k its is called k free. A word is called primitive if it is not of the form w k for k > 1. An integer p is a period of w = a 1... a n if a i = a p+i. The smallest period is often called the period. Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 5 / 13

19 Outline Introduction Fine and Wilf s Theorem Pattern Avoidance Theorems 1 Introduction 2 Fine and Wilf s Theorem Pattern Avoidance Theorems 3 Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 6 / 13

20 Fine and Wilf s Theorem Fine and Wilf s Theorem Pattern Avoidance Theorems Let w be a word of length n. If w has two period p and q and n p + q gcd(p, q), then gcd(p, q) is a period of w. Bound is tight. Consider w = abaab abaab a. It is a word of length 11 and is not power of a single letter. Such words are called Sturmian words. Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 7 / 13

21 Fine and Wilf s Theorem Fine and Wilf s Theorem Pattern Avoidance Theorems Let w be a word of length n. If w has two period p and q and n p + q gcd(p, q), then gcd(p, q) is a period of w. Bound is tight. Consider w = abaab abaab a. It is a word of length 11 and is not power of a single letter. Such words are called Sturmian words. Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 7 / 13

22 Fine and Wilf s Theorem Fine and Wilf s Theorem Pattern Avoidance Theorems Let w be a word of length n. If w has two period p and q and n p + q gcd(p, q), then gcd(p, q) is a period of w. Bound is tight. Consider w = abaab abaab a. It is a word of length 11 and is not power of a single letter. Such words are called Sturmian words. Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 7 / 13

23 Square free and Cube free words Fine and Wilf s Theorem Pattern Avoidance Theorems Find as long as possible (infinite?) word over an n letter alphabet that it is k free. Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 8 / 13

24 Square free and Cube free words Fine and Wilf s Theorem Pattern Avoidance Theorems Find as long as possible (infinite?) word over an n letter alphabet that it is k free. Thue 1906 There exists an infinite 2 + free word over binary alphabet. Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 8 / 13

25 Square free and Cube free words Fine and Wilf s Theorem Pattern Avoidance Theorems Find as long as possible (infinite?) word over an n letter alphabet that it is k free. Thue 1906 There exists an infinite 2 + free word over binary alphabet. Thue 1906 There exists an infinite 2 free word over three alphabet. Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 8 / 13

26 Outline Introduction 1 Introduction 2 Fine and Wilf s Theorem Pattern Avoidance Theorems 3 Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 9 / 13

27 An Example Introduction Consider the following equation All its solution are of the form x = a(ba) n, y = (ba) n b We try to solve one such equation,but over languages Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 10 / 13

28 An Example Introduction Consider the following equation bxyx = yxya All its solution are of the form x = a(ba) n, y = (ba) n b We try to solve one such equation,but over languages Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 10 / 13

29 An Example Introduction Consider the following equation bxyx = yxya All its solution are of the form x = a(ba) n, y = (ba) n b We try to solve one such equation,but over languages Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 10 / 13

30 An Example Introduction Consider the following equation bxyx = yxya All its solution are of the form x = a(ba) n, y = (ba) n b We try to solve one such equation,but over languages Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 10 / 13

31 Problem Statement Conjugacy equation is XZ = ZY Input: X and Y Output: Yes iff there exists a Z. We consider the case when X = {x 1, x 2 } and Y = {y 1, y 2, y 3 } Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 11 / 13

32 Problem Statement Conjugacy equation is XZ = ZY Input: X and Y Output: Yes iff there exists a Z. We consider the case when X = {x 1, x 2 } and Y = {y 1, y 2, y 3 } Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 11 / 13

33 Problem Statement Conjugacy equation is XZ = ZY Input: X and Y Output: Yes iff there exists a Z. We consider the case when X = {x 1, x 2 } and Y = {y 1, y 2, y 3 } Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 11 / 13

34 Problem Statement Conjugacy equation is XZ = ZY Input: X and Y Output: Yes iff there exists a Z. We consider the case when X = {x 1, x 2 } and Y = {y 1, y 2, y 3 } Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 11 / 13

35 Folklore Lemmas Introduction Lemma 1 If p is a primitive word and p 2 = xpy, then either x = 1 or y = 1.In other words, primitive words cannot occur inside their powers in a nontrivial manner. Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 12 / 13

36 Folklore Lemmas Introduction Lemma 1 If p is a primitive word and p 2 = xpy, then either x = 1 or y = 1.In other words, primitive words cannot occur inside their powers in a nontrivial manner. Lemma 2 If xz = zy, then x = (uv) r, y = (vu) r and z = (uv) r u and uv is primitive. Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 12 / 13

37 Folklore Lemmas Introduction Lemma 1 If p is a primitive word and p 2 = xpy, then either x = 1 or y = 1.In other words, primitive words cannot occur inside their powers in a nontrivial manner. Lemma 2 If xz = zy, then x = (uv) r, y = (vu) r and z = (uv) r u and uv is primitive. Lemma 3 If xy = yx, then x = (w) r 1, y = (w) r 2 Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 12 / 13

38 Folklore Lemmas Introduction Lemma 1 If p is a primitive word and p 2 = xpy, then either x = 1 or y = 1.In other words, primitive words cannot occur inside their powers in a nontrivial manner. Lemma 2 If xz = zy, then x = (uv) r, y = (vu) r and z = (uv) r u and uv is primitive. Lemma 3 If xy = yx, then x = (w) r 1, y = (w) r 2 Lemma 4 If XZ = ZY and X commutes then so does Y. Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 12 / 13

39 THANK YOU Kenkireth TIFR (INDIA) A glimpse of combinatorics on words 2 Oct 13 / 13