Definition
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. Most people have a knowledge of factoring large numbers into their decomposition. This idea can be applied to words as well. A factor of a word is a block of consecutive symbols. Using a different example: if the word "summer" is examined, it has the factor "m", since the letter is repeated twice.
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 cycles, and 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. This gives a visual representation of the object.
Read more about this topic: Combinatorics On Words
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