Combinatorics on words

Combinatorics on words is a fairly new field of mathematics, branching from combinatorics, which focuses on the study of words and formal languages. The subject looks at letters or symbols, and 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 the first work was on square-free words by Thue in the early 1900s. 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.

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 analysis, where 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. One of these sets is known by the general public as the alphabet. For example, the word "encyclopedia" is a sequence of symbols in the English alphabet, a 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 set, so 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|. Again looking at the example "encyclopedia", |w| = 12, since encyclopedia has twelve letters. The idea of factoring of large numbers can be applied to words, where a factor of a word is a block of consecutive symbols. Thus, "cyclop" is a factor of "encyclopedia".

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