DOCSLIB.ORG

Axel Thue's Papers on Repetitions in Words: a Translation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Axel Thue's Papers on Repetitions in Words: a Translation Axel Thues pap ers on rep etitions in words a translation J Berstel LITP Institut Blaise Pascal Universite Pierre et Marie Curie Paris France July Contents Preliminaries Notation Co des and enco dings The ThueMorse sequence Symb olic dynamical systems Thues First Pap er Ab out innite sequences of symb ols x x Thues Second Pap er On the relative p osition of equal parts in certain sequences of symb ols Intro ductory Remarks Sequences over two symb ols Sequences over three symb ols First Case aca and bcb are missing Second Case aba and aca are missing Third Case aba and bab are missing Irreducible words over four letters Irreducible words over more than four letters Notes Squarefree morphisms Overlapfree words Avoidable patterns Intro duction In a series of four pap ers which app eared during the p erio d Axel Thue considered several combinatorial problems which arise in the study of sequences of symb ols Two of these pap ers deal with word problems for nitely presented semigroups these pap ers contain the denition of what is now called a Thue system He was able to solve the word problem in sp ecial cases It was only in that the general case was shown to b e unsolvable indep endently by E L Post and A A Markov The other two pap ers deal with rep etitions in nite and innite words Perhaps b ecause these pap ers were published in a journal with restricted avail ability this is guessed by G A Hedlund this work of Thue was widely ignored during a long time and consequently some of his results have b een rediscovered again and again Axel Thues pap ers on sequences are now more easily accessible since they are included in the Selected Pap ers which were edited in It is the purp ose of the present text to give a translation of Axel Thues pap ers on rep etitions in sequences b oth in more recent terminology and in relation with new results and directions of research It app ears that there is a noticeable dierence b oth in style and in amount of results b etween the pap er pages and the pap er pages The rst of these pap ers mainly contains the construction of an innite square free word over three letters Thue gives also an innite squarefree word over four letters obtained by what is now called an iterated morphism whilst the three letter word is constructed in a slightly more complicated way a uniform tagsystem in the terminology of Cobham The second pap er attacks the more general problem of what Thue calls irre ducible words He devotes sp ecial attention to the case of two and three letters In particular he intro duces what is now called the ThueMorse sequence and shows that all twosided innite overlapfree words are derived from this se quence There are several asp ects he did not consider rst many combinatorial prop erties of the ThueMorse sequence such as the numb er of factors the recur rence index and so on were only investigated by M Morse or later next Intro duction the characterization of all onesided innite overlapfree words which is much more dicult than that of twosided words was only given later by Fife However Thue gives a complete description of circular overlapfree words Axel Thues investigation of squarefree words over three letters is even more detailed He gives in this pap er another construction of an innite squarefree word by iterated morphism and then initiates in a pages development a tentative to describ e all squarefree words over three letters He observes that every innite squarefree word is an innite pro duct of words chosen in a set of six words and classies those innite squarefree words that are pro ducts of four among these six words His classication he observes is similar b oth in statement and in pro of technique to what is found in diophantine equations the solutions are parametrized by some variables which are easier to manage This text is organized as follows in the rst chapter we give some preliminary denitions and notation We intro duce the notions of squarefree overlapfree words avoidable pattern morphisms and co des These are useful to present Thues results in a somewhat more concise manner As an example we give some combinatorial prop erties of the ThueMorse sequence The two following chapters contain a translation of Thues pap ers We have tried to formulate Thues results as faithfully as p ossible For the pro ofs some easy parts have b een simplied and more frequently some dicult steps have b een develop ed In these chapters fo otnotes only concern technical details A longer chapter of notes contains more general remarks and developments b oth ab out the contents of Thues pap ers and ab out the actual state of the art Chapter Preliminaries In this preliminary chapter we rst intro duce some denitions and notation and then present the socalled ThueMorse sequence and some of its prop erties Notation An alphabet is a nite set of symbols or letters A word over some alphab et A is a nite sequence of elements in A The length of a word w is denoted by jw j The empty word of length is denoted by We denote by alphw the set of letters that o ccur at least once in the word w An innite word is a mapping from N into A and a twosided innite word is a mapping from Zinto A A circular word or necklace is the equivalence class of a nite word under conjugacy or circular p ermutation We shall write u w if u and w dene the same circular word Sometimes we identify a circular word with one of its representatives A factor of a word w is any word u that o ccurs in w i e such that there exist words x y with w xuy A square is a nonempty word of the form uu A word is squarefree if none of its factors is a square Similarly an overlap is a word of the form xuxux where x is nonempty The terminology is justied by the fact that xux has two o ccurrences in xuxux one as a prex initial factor one as a sux nal factor and that these o ccurrences have a common part the central x As b efore a word is overlapfree if none of its factors is an overlap The reversal of a word u a a where a a are letters is the word n n u a a If u u then u is a palindrome The reversal of an innite word n to the right is an innite word to the left The set of words over A is the free monoid generated by A and is denoted by A The set of nonempty words over A is denoted by A It is the free semigroup generated by A A function h A B is a morphism if huv huhv for Co des and enco dings all words u v If jhw j jw j for all words w then h is nonerasing or length increasing It is equivalent to say that hw for w If there is a letter a n n such that ha starts with the letter a then h a starts with the word h a n for all n If the set of words fh a j n g is innite the morphism denes n a unique innite word say x by the requirement that all h a are prexes of x The word x is said to b e obtained by iterating h on a and is called a morphic word Sometimes x is also denoted by h a Clearly x is a xed p oint of h The ThueMorse sequence of section is an example of a morphic word A morphism h A B easily extends to onesided innite words If x a a a is an innite word then hx ha ha ha The n n resulting word is innite i the set of indices n such that ha is innite n This holds in particular if h is nonerasing The extension to twosided innite words is similar The only ambiguity is in the convention adopted to x the origin of the image We agree that any origin is convenient In other words we consider insofar as homomorphic images are concerned the equivalence class under the shift operator T that is dened by T xn xn If u is a nite juj word then the innite p erio dic word u uuu veries u T u Co des and enco dings A code over A is a set X of nonempty words such that each word over A admits at most one factorization as a pro duct of words in X In other words for all n m x x y y X n m x x y y n m and x y i n n m i i It is equivalent to say that the submonoid X generated by X is free and that X is its base A set X is prex if no word in X is a prex of any other word in X thus x xu X implies u Sux sets are dened symmetrically Prex and sux sets are co des A biprex co de is a co de that is b oth prex and sux An encoding is a morphism h A B that is injective If h is an enco ding then the set X hA is a co de Conversely if X is a co de over an alphab et B then an enco ding of X is obtained by taking a bijection h from an alphab et A onto X This extends to an injective morphism from A into B It is convenient to implicitl y transfer terminology b etween co des and enco dings Thus we may sp eak ab out prex enco dings or ab out comp osition of co des Several sp ecial prop erties of co des are useful and will b e intro duced when they are needed Preliminaries The ThueMorse sequence In this section we recall some basic prop erties concerning the ThueMorse se quence Other prop erties and pro ofs can b e found in Lothaire and Salo maa and of course in Thues second pap er Let A fa bg b e a two letter alphab et Consider the morphism from the free monoid A into itself dened by a ab b ba Setting for n n n u
Load more

Recommended publications
  • A Generalization of Square-Free Strings a GENERALIZATION of SQUARE-FREE STRINGS
    A Generalization of Square-free Strings A GENERALIZATION OF SQUARE-FREE STRINGS BY NEERJA MHASKAR, B.Tech., M.S. a thesis submitted to the department of Computing & Software and the school of graduate studies of mcmaster university in partial fulfillment of the requirements for the degree of Doctor of Philosophy c Copyright by Neerja Mhaskar, August, 2016 All Rights Reserved Doctor of Philosophy (2016) McMaster University (Dept. of Computing & Software) Hamilton, Ontario, Canada TITLE: A Generalization of Square-free Strings AUTHOR: Neerja Mhaskar B.Tech., (Mechanical Engineering) Jawaharlal Nehru Technological University Hyderabad, India M.S., (Engineering Science) Louisiana State University, Baton Rouge, USA SUPERVISOR: Professor Michael Soltys CO-SUPERVISOR: Professor Ryszard Janicki NUMBER OF PAGES: xiii, 116 ii To my family Abstract Our research is in the general area of String Algorithms and Combinatorics on Words. Specifically, we study a generalization of square-free strings, shuffle properties of strings, and formalizing the reasoning about finite strings. The existence of infinitely long square-free strings (strings with no adjacent re- peating word blocks) over a three (or more) letter finite set (referred to as Alphabet) is a well-established result. A natural generalization of this problem is that only sub- sets of the alphabet with predefined cardinality are available, while selecting symbols of the square-free string. This problem has been studied by several authors, and the lowest possible bound on the cardinality of the subset given is four. The problem remains open for subset size three and we investigate this question. We show that square-free strings exist in several specialized cases of the problem and propose ap- proaches to solve the problem, ranging from patterns in strings to Proof Complexity.
    [Show full text]
  • Some Recent Results on Squarefree Words
    Some recent results on squarefree words by Jean Berstel Universit~ Pierre et Marie Curie and L.I.T.P Paris "F~r die Entwicklung der logischen Wissenschaften wird es, ohne ROcksicht auf etwaige Anwendungen, yon Bedeutung sein, ausgedehnte Felder f~r Spekulation Ober schwierige Probleme zu finden." Axel Thue, 1912. I. Introduction. When Axel Thue wrote these lines in the introduction to his 1912 paper on squarefree words, he certainly did not feel as a theoretical computer scientist. During the past seventy years, there was an increasing interest in squarefree words and more generally in repetitions in words. However, A. Thue's sentence seems still to hold : in some sense, he said that there is no reason to study squarefree words, excepted that it's a difficult question, and that it is of primary importance to investigate new domains. Seventy years later, these questions are no longer new, and one may ask if squarefree words served already. First, we observe that infinite squarefree, overlap-free or cube-free words indeed served as examples or counter-examples in several, quite different domains. In symbolic dynamics, they were introduced by Morse in 1921 [36] . Another use is in group theory, where an infinite square-free word is one (of the numerous) steps in disproving the Burnside conjecture (see Adjan[2]). Closer to computer science is Morse and Hedlund's interpretatio~ in relation with chess [37]. We also mention applications to formal language theory : Brzozowsky, K. Culik II and Gabriellian [7] use squarefree words in connection with noncounting languages, J. Goldstine uses the Morse sequence to show that a property of some family of languages [22].
    [Show full text]
  • Combinatorics on Words: Christoffel Words and Repetitions in Words
    Combinatorics on Words: Christoffel Words and Repetitions in Words Jean Berstel Aaron Lauve Christophe Reutenauer Franco Saliola 2008 Ce livre est d´edi´e`ala m´emoire de Pierre Leroux (1942–2008). Contents I Christoffel Words 1 1 Christoffel Words 3 1.1 Geometric definition ....................... 3 1.2 Cayley graph definition ..................... 6 2 Christoffel Morphisms 9 2.1 Christoffel morphisms ...................... 9 2.2 Generators ............................ 15 3 Standard Factorization 19 3.1 The standard factorization .................... 19 3.2 The Christoffel tree ........................ 23 4 Palindromization 27 4.1 Christoffel words and palindromes ............... 27 4.2 Palindromic closures ....................... 28 4.3 Palindromic characterization .................. 36 5 Primitive Elements in the Free Group F2 41 5.1 Positive primitive elements of the free group .......... 41 5.2 Positive primitive characterization ............... 43 6 Characterizations 47 6.1 The Burrows–Wheeler transform ................ 47 6.2 Balanced1 Lyndon words ..................... 50 6.3 Balanced2 Lyndon words ..................... 50 6.4 Circular words .......................... 51 6.5 Periodic phenomena ....................... 54 i 7 Continued Fractions 57 7.1 Continued fractions ........................ 57 7.2 Continued fractions and Christoffel words ........... 58 7.3 The Stern–Brocot tree ...................... 62 8 The Theory of Markoff Numbers 67 8.1 Minima of quadratic forms .................... 67 8.2 Markoff numbers ......................... 69 8.3 Markoff’s condition ........................ 71 8.4 Proof of Markoff’s theorem ................... 75 II Repetitions in Words 81 1 The Thue–Morse Word 83 1.1 The Thue–Morse word ...................... 83 1.2 The Thue–Morse morphism ................... 84 1.3 The Tarry-Escott problem .................... 86 1.4 Magic squares ........................... 90 2 Combinatorics of the Thue–Morse Word 93 2.1 Automatic sequences ......................
    [Show full text]
  • CRM Monograph Series ; V
    Volume 27 C CRM R MONOGRAPH M SERIES Centre de Recherches Mathématiques Montréal Combinatorics on Words Christoffel Words and Repetitions in Words Jean Berstel Aaron Lauve Christophe Reutenauer Franco V. Saliola American Mathematical Society Combinatorics on Words Christoffel Words and Repetitions in Words https://doi.org/10.1090/crmm/027 Volume 27 C CRM R MONOGRAPH M SERIES Centre de Recherches Mathématiques Montréal Combinatorics on Words Christoffel Words and Repetitions in Words Jean Berstel Aaron Lauve Christophe Reutenauer Franco V. Saliola The Centre de Recherches Mathématiques (CRM) of the Université de Montréal was created in 1968 to promote research in pure and applied mathematics and related disciplines. Among its activities are special theme years, summer schools, workshops, postdoctoral programs, and publishing. The CRM is supported by the Université de Montréal, the Province of Québec (FQRNT), and the Natural Sciences and Engineering Research Council of Canada. It is affiliated with the Institut des Sciences Mathématiques (ISM) of Montréal, whose constituent members are Concordia University, McGill University, the Université de Montréal, the Université du Québec à Montréal, and the E´ cole Polytechnique. The CRM may be reached on the Web at www.crm.umontreal.ca. American Mathematical Society Providence, Rhode Island USA The production of this volume was supported in part by the Fonds Qu´eb´ecois de la Recherche sur la Nature et les Technologies (FQRNT) and the Natural Sciences and Engineering Research Council of Canada (NSERC). 2000 Mathematics Subject Classification.Primary68R15; Secondary 37B10, 11J70, 68W40. For additional information and updates on this book, visit www.ams.org/bookpages/crmm-27 Library of Congress Cataloging-in-Publication Data Combinatorics on words : Christoffel words and repetitions in words / Jean Berstel ..
    [Show full text]
  • Grasshopper Avoidance of Patterns
    Grasshopper avoidance of patterns Micha lD¸ebski Urszula Pastwa Krzysztof W¸esek Faculty of Mathematics and Information Science Warsaw University of Technology Warsaw, Poland [email protected], [email protected], [email protected] Submitted: Jun 10, 2016; Accepted: Oct 15, 2016; Published: Oct 28, 2016 Mathematics Subject Classifications: 68R15 Abstract Motivated by a geometrical Thue-type problem, we introduce a new variant of the classical pattern avoidance in words, where jumping over a letter in the pattern occurrence is allowed. We say that pattern p 2 E+ occurs with jumps in a word + ∗ ∗ w = a1a2 : : : ak 2 A , if there exist a non-erasing morphism f from E to A and a sequence (i1; i2; : : : ; il) satisfying ij+1 2 fij + 1; ij + 2g for j = 1; 2; : : : ; l − 1, such that f(p) = ai1 ai2 : : : ail : For example, a pattern xx occurs with jumps in a word abdcadbc (for x 7! abc). A pattern p is grasshopper k-avoidable if there exists an alphabet A of k elements, such that there exist arbitrarily long words over A in which p does not occur with jumps. The minimal such k is the grasshopper avoidability index of p. It appears that this notion is related to two other problems: pattern avoidance on graphs and pattern-free colorings of the Euclidean plane. In particular, we show that a sequence avoiding a pattern p with jumps can be a tool 2 to construct a line p-free coloring of R . In our work, we determine the grasshopper avoidability index of patterns αn for all n except n = 5.
    [Show full text]
  • The Complexity of Unavoidable Word Patterns
    The complexity of unavoidable word patterns by PAUL VAN DER MERWE SAUER submitted in accordance with the requirements of DOCTOR OF PHILOSOPHY in the subject OPERATIONS RESEARCH at the UNIVERSITY OF SOUTH AFRICA SUPERVISOR: PROF W L FOUCHÉ CO-SUPERVISOR: PROF P H POTGIETER December 30, 2019 2 Summary The avoidability, or unavoidability of patterns in words over finite alphabets has been studied extensively. The word α over a finite set A is said to be unavoidable for an infinite set B+ of nonempty words over a finite set B if, for all but finitely + + + many elements w of B , there exists a semigroup morphism φ ∶ A → B such that φ(α) is a factor of w. In this treatise, we start by presenting a historical background of results that are related to unavoidability. We present and discuss the most important theorems surrounding unavoidability in detail. We present various complexity-related properties of unavoidable words. For words that are unavoidable, we provide a constructive upper bound to the lengths of words that avoid them. In particular, for a pattern α of length n over an alphabet of size r, we give a concrete function N(n; r) such that no word of length N(n; r) over the alphabet of size r avoids α. A natural subsequent question is how many unavoidable words there are. We show that the fraction of words that are unavoidable drops exponentially fast in the length of the word. This allows us to calculate an upper bound on the number of unavoidable patterns for any given finite alphabet.
    [Show full text]
  • Relations on Words 3
    RELATIONS ON WORDS MICHEL RIGO Abstract. In the first part of this survey, we present classical notions arising in combi- natorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our presentation tries to set up a unified framework with respect to a given binary relation. In the second part, we mainly focus on abelian equivalence, k-abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and M-equivalence. In particular, some new refinements of abelian equivalence are introduced. 1. Introduction This paper follows and complements the talk I gave during the conference on Automatic Sequences, Number Theory, and Aperiodic Order held in Delft in October 2015. The aim is to survey various concepts arising in combinatorics on words and present them in a unified and general framework. In Section 3, relatively to a given binary relation over A∗, we define the growth function of a language, the complexity function of an infinite word and the notions of avoidable patterns, periodicity and uniform recurrence. These notions are usually first introduced for the restrictive case of equality of factors, e.g., the complexity function counts the number of factors of length n occurring in a given infinite word but we could count them up to rearrangement of the letters. In the second part of the paper, we review classical binary relations on words where these concepts may be applied. In Section 4, we consider abelian equivalence, then its extension to k-abelian equivalence is presented in Section 5. We pursue in Section 6 with binomial coefficients of words and various equivalence relations that can be associated with.
    [Show full text]
  • Patterns, Pattern Avoidance and Graphs on Words
    Patterns, Pattern Avoidance, and Graphs on Words Thesis submitted in partial fulfilment of the requirements for the award of the degree of Doctor of Philosophy in Computer Science and Engineering by Mrityunjay Singh Under the supervision of Dr. Benny George Kenkireth Department of Computer Science and Engineering Indian Institute of Technology Guwahati Guwahati - 781039 Assam India MAY, 2019 Copyright c Mrityunjay Singh 2019. All Rights Reserved. I dedicate this thesis to my late grand mother. Acknowledgements First and foremost, I thank God Almighty. I would like to take this opportunity to acknowl- edge the support and help of my guide Dr. Benny George Kenkireth. He has motivated, supported, and guided me throughout this work. This work has been a really nice learning experience for me. He has taught me how to communicate mathematics. I would also like to thank my doctoral committee members Prof G. Sajith, Prof. Kalpesh Kapoor, Dr. Deepan- jan Kesh and Prof. K.V. Krishna for their guidance through out the duration of my doctoral studies. I would like to thank to the professors P.A.S. Krishna, Shyamsree Upadhyay, Bhaba Kumar Sharma, Rupam Barman, Bikash Bhattacharya, Deepanjan Kesh, Sushant Karmakar, Pinaki Mitra, Purandar Bhaduri, V. Vijaya Saradhi, Sukumar Nandi, H. Ramesh and T. Venketesh for the various courses they taught me during my stay at IITG. I would also like to thank professors John Tsitsiklis, Vittal Rao, Fedrico Ardila, Arindama Singh, Somenath Biswas and Tim Roughgarden for the various online courses taught by them. I thank the whole IITG fraternity for all the different ways in which they have knowingly and unknowingly help me.
    [Show full text]
  • Some Recent Results on Squarefree Words Publications Du Département De Mathématiques De Lyon, 1985, Fascicule 2B « Compte Rendu Des Journées Infinitistes », , P
    PUBLICATIONS DU DÉPARTEMENT DE MATHÉMATIQUES DE LYON JEAN BERSTEL Some Recent Results on Squarefree Words Publications du Département de Mathématiques de Lyon, 1985, fascicule 2B « Compte rendu des journées infinitistes », , p. 21-36 <http://www.numdam.org/item?id=PDML_1985___2B_21_0> © Université de Lyon, 1985, tous droits réservés. L’accès aux archives de la série « Publications du Département de mathématiques de Lyon » im- plique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ SOME RECENT RESULTS ON SQUAREFREE WORDS by Jean BERSTEL <1 > "Fur die Entwicklung der logischen Wissenschaften wird es, oh ne Rucks icht ouf etwaige Anwendungen, von Bedeutung sein, ausgedehnte Felder fur Spekulation ùber schwierige Problème zu finden. 11 Axel Thue, 1912. 1. Introduction. When Axel Thue wrote these lines in the introduction to his 1912 paper on squarefree words, he certainly did not feel as a theoretical computer scientist. During the past seventy years, there was an increasing interest in squarefree words and more generally in repetitions in words. However, A. Thuefs sentence seems still to hold : in some sense, he said that there is no reason to study squarefree words, excepted that it1s a difficult question, and that it is of primary importance to investigate new domains. Seventy years later, these questions are no longer new, and one may ask if squarefree words served already.
    [Show full text]
  • Avoidable Formulas in Combinatorics on Words
    University of California Los Angeles Avoidable Formulas in Combinatorics on Words A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Ronald James Clark 2001 The dissertation of Ronald James Clark is approved. D. Stott Parker Robert Brown Bruce Rothschild Kirby Baker, Committee Chair University of California, Los Angeles 2001 ii For Azusa, partner-in-crime and partner-in-life iii Table of Contents 1 Preliminaries ............................... 1 1.1 Introduction . 1 1.2 Basic Definitions . 2 1.3 Formulas and Divisibility . 3 1.4 Avoidability . 6 1.5 A Known Algorithm to Determine Avoidability . 10 1.6 Finding the Index of Avoidability . 12 1.7 Locked Formulas . 18 2 Avoidance Bases ............................. 20 2.1 Introduction . 20 2.2 Avoidance Bases . 20 2.3 Finding Avoidance Bases . 23 2.4 Avoidance Bases on Small Alphabets . 27 2.5 Formulas on Three Letters Are 4-Avoidable . 28 3 Minimally Locked Formulas ...................... 44 3.1 Introduction . 44 3.2 Indices of MLF’s on Small Alphabets . 45 3.3 Growth Rates . 48 4 Index 5 Formulas ............................. 53 iv 4.1 Introduction . 53 4.2 The Formula ρ Is 4-Unavoidable . 54 4.3 5-Special Formulas . 54 4.4 5-Special Formulas Are 5-Avoidable . 56 4.5 Index 6? . 65 A Cubefree Words on Two Letters ................... 66 A.1 Introduction . 66 A.2 Fixed Points of Endomorphisms . 67 A.3 A Bound for the Density Function . 71 A.4 Full Density . 73 A.5 Growth Rates . 77 References ................................... 78 v List of Tables 1.1 Cassaigne’s Backtracking Algorithm .
    [Show full text]
  • Appendix a Useful Notions and Facts
    Appendix A Useful notions and facts In this appendix we introduce most of the statistics on permutations and words ap- pearing in this book, as well as several well-known sequences of numbers and certain lattice paths. Also, we state in Table A.7 several classes of restricted permutations together with references and initial values of the corresponding sequences. A.1 Statistics on permutations and words Definition A.1.1. In a permutation π = π1π2 : : : πn, πi is a left-to-right maximum (resp., left-to-right minimum) if πi ≥ πj (resp., πi ≤ πj) for 1 ≤ j ≤ i. Also, πi is a right-to-left maximum (resp., right-to-left minimum) if πi ≥ πj (resp., πi ≤ πj) for i ≤ j ≤ n. Example A.1.2. If π = 3152647 then the left-to-right maxima are 3, 5, 6 and 7, the left-to-right minima are 3 and 1, the right-to-left maximum is 7, and the right-to-left minima are 7, 4, 2 and 1. In Table A.1 we define all permutation statistics appearing in the book. The list of statistics is essentially taken from [259]. Note that many of the statistics defined make sense for words, though not all. For example, the statistic head :i, the position of the smallest letter, is not defined for words as we may have several instances of the smallest letter. The expression π1 appearing in Table A.1 means the permutation beginning with π and ending with a letter that is larger that any letter in π. Remark A.1.3.
    [Show full text]
  • Pattern Occurrence Statistics and Applications to the Ramsey Theory
    Pattern occurrence statistics and applications to the Ramsey theory of unavoidable patterns∗ Jim Tao1 October 29, 2018 Abstract As suggested by Currie, we apply the probabilistic method to problems regarding pattern avoidance. Using techniques from analytic combinatorics, we calculate asymptotic pattern oc- currence statistics and use them in conjunction with the probabilistic method to establish new results about the Ramsey theory of unavoidable patterns in the full word case (both nonabelian sense and abelian sense) and in the partial word case. Keywords: Combinatorics on words; Partial words; Unavoidable patterns; Abelian patterns; Probabilistic method; Analytic combinatorics; Ramsey theory. 1 Introduction In [10], Currie reviews results and formulates a large number of open problems concerning pattern avoidance as well as an abelian variation of it. Given a pattern p over an alphabet and a word w V over an alphabet , we say that w encounters p if there exists a nonerasing morphism h : ∗ ∗ A V →A such that h(p) is a factor of w; otherwise w avoids p. In other words, w encounters p = p p , 1 · · · n where p ,...,p , if w contains u u as a factor, where u , . , u are nonempty words in 1 n ∈ V 1 · · · n 1 n ∗ satisfying u = u whenever p = p . On the other hand, w encounters p = p p in the A i j i j 1 · · · n abelian sense if w contains u u as a factor, where u can be obtained from u by rearranging 1 · · · n i j letters whenever pi = pj; otherwise w avoids p in the abelian sense.
    [Show full text]