Combinatorics

Information Technology; Algorithms; System Analysis

Combinatorics is a branch of mathematics that is concerned with sets of objects that meet certain conditions. In computer science, combinatorics is used to study algorithms, which are sets of steps, or rules, devised to address a certain problem.

Combinatorics is a branch of mathematics that studies counting methods and combinations, permutations, and arrangements of sets of objects. For instance, given a set of fifteen different objects, combinatorics studies equations that determine how many different sets of five can be created from the original set of fifteen. The study of combinatorics is crucial to the study of algorithms. Algorithms are sets of rules, steps, or processes that are linked together to address a certain problem.

Combinatorics is often called the “science of counting.” It focuses on the properties of finite sets of objects, which do not have infinite numbers of objects and so are theoretically countable. The process of describing or counting all of the items in a specific set is called “enumeration.” Combinatorics also includes the study of combinations, a process of selecting items from a set when the order of selection does not matter. Finally, combinatorics also studies permutations. Permutations involve selecting or arranging items in a list, when the order of arrangement is important. Combinatorics also studies the relationships between objects organized into sets in various ways.

There are numerous subfields of combinatorics used to study sets of objects in different ways. Enumerative combinatorics is the most basic branch of the field. It can be described as the study of counting methods used to derive the number of objects in a given set. By contrast, analytic combinatorics is a subfield of enumerative combinatorics. It deals with predicting the properties of large sets of objects, using quantitative analysis. All combinatorics analysis requires detailed knowledge of calculus. Many subfields make extensive use of probability theory and predictive analysis.

There are many different applications for combinatorics in analytic mathematics, engineering, physics, and computer science. Among the most familiar basic examples of combinatorics is the popular game sudoku. The game challenges players to fill in the blanks in a “magic square” diagram with specific column and row values. Sudoku puzzles are an example of combinatorial design, which is a branch of combinatorics that studies arrangements of objects that have symmetry or mathematical/geometric balance between the elements.

Combinatorics is also crucial to graph theory. Graph theory is a field of mathematics that deals with graphs, or representations of objects in space. Graphs are used in geometry, computer science, and other fields to model relationships between objects. In computer science, for instance, graphs are typically used to model computer networks, computational flow, and the structure of links within websites. Combinatorics is used to study the enumeration of graphs. This can be seen as counting the number of different possible graphs that can be used for a certain application or model.

In computer science, combinatorics is used in the creation and analysis of algorithms. Algorithms are sets of instructions, computing steps, or other processes that are linked together to address a certain computational problem. As algorithms are essentially sets, the steps within algorithms can be studied using combinatorial analysis, such as enumeration or permutation. As combinatorics can help researchers to find more efficient arrangements of objects, or sets, combinatorial analysis is often used to test and assess the efficiency of computer algorithms. Combinatorics is also key to the design of sorting algorithms. Sorting algorithms allow computers to sort objects (often pieces of data, web pages, or other informational elements), which is a necessary step in creating effective algorithms for web searching.

Combinatorics is also used in coding theory, the study of codes and their associated properties and characteristics. Codes are used for applications, including cryptography, compressing or translating data, and correcting errors in mathematical, electrical, and information systems. Coding theory emerged from combinatorial analysis. These two branches of mathematics are distinct but share theories and techniques.

—Micah L. Issit