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Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures.It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science.
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For any pair of positive integers n and k, the number of k-tuples of non-negative integers whose sum is n is equal to the number of multisets of size k − 1 taken from a set of size n + 1, or equivalently, the number of multisets of size n taken from a set of size k, and is given by
Combinatorial designs date to antiquity, with the Lo Shu Square being an early magic square.One of the earliest datable application of combinatorial design is found in India in the book Brhat Samhita by Varahamihira, written around 587 AD, for the purpose of making perfumes using 4 substances selected from 16 different substances using a magic square.
The Fano matroid, derived from the Fano plane.Matroids are one of many kinds of objects studied in algebraic combinatorics. Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra.
Extremal combinatorics is a field of combinatorics, which is itself a part of mathematics.Extremal combinatorics studies how large or how small a collection of finite objects (numbers, graphs, vectors, sets, etc.) can be, if it has to satisfy certain restrictions.
Geometric combinatorics is a branch of mathematics in general and combinatorics in particular. It includes a number of subareas such as polyhedral combinatorics (the study of faces of convex polyhedra), convex geometry (the study of convex sets, in particular combinatorics of their intersections), and discrete geometry, which in turn has many applications to computational geometry.