This is the third book in the Lothaire’s series, following the volumes “ Combinatorics on Words” and “Algebraic Combinatorics on Words” already published. A series of important applications of combinatorics on words has words. Lothaire’s “Combinatorics on Words” appeared in its first printing in. Combinatorics on words, or finite sequences, is a field which grew simultaneously within disparate branches of mathematics such as group theory and.

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Combinatorics on words is a fairly new field of mathematicsbranching from combinatoricswhich focuses on the study of words and formal languages.

The subject looks at letters or symbolsand the sequences they form. Combinatorics on words affects various areas of mathematical study, including algebra and computer science. There have been a wide range of contributions to the field. Some of combinatorids first work was on square-free words by Thue in the early s. He and colleagues observed patterns within words and tried to explain them. As time went on, combinatorics on words became useful in the study of algorithms and coding.

It led to developments in abstract algebra and answering open questions.

### M. Lothaire – Wikipedia

Combinatorics is an area of discrete mathematics. Discrete mathematics is the study of countable structures.

These objects have a definite beginning and end. The study of enumerable objects is the opposite of disciplines such as analysiswhere calculus and infinite structures are studied. Combinatorics studies how to count these objects using various representation. Combinatorics on words is a recent development in this field, which focuses on the study of words and formal languages. A formal language is any set of symbols and combinations of symbols that people use to communicate information.

Some terminology relevant to the study of words should first be explained. First and foremost, a word is basically a sequence of symbols, or letters, in a finite set. For example, the word “encyclopedia” is a sequence of symbols in the English alphabeta finite set of twenty-six letters. Since a word can be described as a sequence, other basic mathematical descriptions can be applied.

The alphabet is a setso as one would expect, the empty set is a subset. In other words, there exists a unique word of length zero. The length of the word is defined by the number of symbols that make up the sequence, and is denoted by w. The idea of factoring of large numbers can be applied to words, where a factor of a word is a block of consecutive symbols. In addition to examining sequences in themselves, another area to consider of combinatorics on words is how they can be represented visually.

In mathematics various structures are used to encode data. A common structure used in combinatorics is referred to as a tree structure. A tree structure is a graph where the vertices are connected by one line, called a path or edge. These trees may or may not contain cyclesand may or may not be complete. It is possible to encode a word, since a word is constructed by symbols, and encode the data by using a tree. The first books on combinatorics on words that summarize the origins of the subject were written by a group of mathematicians that collectively went by the name of M.

## M. Lothaire

Their first book was published inwhen combinatorics on words became more widespread. A main contributor to the development of combinatorics on words was Axel Thue — ; he researched repetition.

Thue’s main contribution was the proof of the existence of infinite square-free words. Square-free words do lothare have adjacent repeated factors. Thue proves his conjecture on the existence of infinite square-free words by using substitutions. A substitution is a way to take a symbol and replace it with a word. He uses this technique to describe his other contribution, the Thue—Morse sequenceor Thue—Morse word. Combinatorkcs wrote two papers on square-free words, the second of which was on the Thue—Morse word.

Marston Morse is included in the name because he discovered the same result as Thue did, yet they worked independently. Thue also proved the existence of an overlap-free word. An overlap-free word is when, for two symbols x and y, the pattern xyxyx does not exist within the word.

He continues in his second paper to prove a relationship between combinatorice overlap-free words and square-free words. He takes overlap-free words that are created using two different letters, and demonstrates how they can be transformed into square-free words of three letters using substitution.

As was previously described, words are studied by examining the sequences made by the symbols. Patterns are found, and they are able to be described mathematically. Patterns can combintorics either avoidable patterns, or unavoidable.

A significant contributor to the work of unavoidable patternsor regularities, was Frank Ramsey in Other contributors to the study of unavoidable patterns include van der Waerden. His theorem states that if the positive integers are partitioned into k classes, then there exists a class c such that c contains an arithmetic progression of some unknown length.

An arithmetic progression is a sequence of numbers in which the difference between adjacent numbers remains constant. When lothairre unavoidable patterns sesquipowers are also studied.

For some combinatorice x,y,z, a sesquipower is of the form x, xyx, xyxzxyx, This is another pattern such as square-free, or unavoidable patterns. In addition, Zimin proved that sesquipowers are all unavoidable. Whether the entire pattern shows up, or only some piece of the sesquipower shows up repetitively, it is not possible to avoid wirds. Necklaces are constructed from words of circular sequences. They combjnatorics most frequently used in music and astronomy. A de Bruijn necklace contains factors made of words of length n over a certain number of letters.

The words appear only once in the necklace. InBaudot developed the code that would eventually take the place of Morse code by applying the theory of binary de Bruijn necklaces. The problem continued from Sainte-Marie to Martin inwho began looking at algorithms wodrs make words of the de Bruijn structure. It was then worked on by Combinaatorics in Possibly the most applied result in combinatorics on words is the Chomsky hierarchy, [ verification needed ] developed by Noam Chomsky.

He studied formal language in the s.

He disregards the actual meaning of the word, does not consider certain factors such as frequency and context, and applies patterns of short terms to all length terms. The basic idea of Chomsky’s work is to divide language into four levels, or the language hierarchy.

The four levels are: While his work grew out of combinatorics on lotuaire, it drastically affected other lothair, especially computer science. There exist several equivalent definitions of Sturmian words. A Lyndon word is a word over a given alphabet that is written in ln simplest and most ordered form out of its respective conjugacy class. Further, there exists a theorem by Chen, Fox, and Lyndon,othaire states any word has a unique factorization of Lyndon words, where the factorization words are non-increasing.

Due to this property, Lyndon words are used to study algebraspecifically group theory. They form the basis for the idea of commutators. Cobham contributed work relating Prouhet’s work with finite automata. A mathematical graph is made of edges and nodes.

With finite automata, the edges are labeled with a letter in an alphabet.

## Combinatorics on words

To use the graph, one starts at a node and travels along the edges to reach a final node. The path taken along the graph forms the word. It is a finite graph because there are a countable number of nodes and edges, and only one path connects two distinct nodes. Gauss codescreated by Carl Friedrich Gauss inare developed from graphs. Specifically, a closed curve on a plane is combunatorics. If the curve only crosses over itself a finite number of times, then one labels the intersections with a letter from the alphabet used.

Traveling along the curve, the word is determined by recording each letter as an intersection is passed. Gauss noticed that the distance between when the same symbol shows up in a word is an even integer. Walther Franz Anton von Dyck began the work of combinatorics on words in group theory by his pothaire work in and He began by using words as group elements. Lagrange also contributed in with his work on permutation groups.

One aspect of combinatorics on words studied in group theory is reduced words. Nielsen transformations were also developed. For a set of elements of a free groupa Nielsen transformation is achieved by three transformations; replacing an element with its inverse, replacing an element with the product of itself and another element, and eliminating any element equal to 1.

By applying these transformations Nielsen reduced sets are formed. A reduced set means no element can be multiplied by other elements to cancel out completely. There oh also connections with Nielsen transformations with Sturmian words.

One problem considered in the study of combinatorics on words in group theory is the following: Post and Markov studied this problem and determined it undecidable. Undecidable means the theory cannot be proved. The Burnside question was proved using the existence of an infinite cube-free word. Many word problems are undecidable based on the Post correspondence problem. Post proved that this problem is undecidable; consequently, any word problem that can be reduced to this basic problem is likewise undecidable.