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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.
In combinatorics, stars and bars (also called "sticks and stones", [1] "balls and bars", [2] and "dots and dividers" [3]) is a graphical aid for deriving certain combinatorial theorems.
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.
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.
A combinatorial explosion can also occur in some puzzles played on a grid, such as Sudoku. [2] A Sudoku is a type of Latin square with the additional property that each element occurs exactly once in sub-sections of size √ n × √ n (called boxes).
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.
In mathematics, a composition of an integer n is a way of writing n as the sum of a sequence of (strictly) positive integers.Two sequences that differ in the order of their terms define different compositions of their sum, while they are considered to define the same integer partition of that number.
Possible patterns of bracelets of length n corresponding to the k-th integer partition (set partitions up to rotation and reflection)For a given set of n beads, all distinct, the number of distinct necklaces made from these beads, counting rotated necklaces as the same, is n! / n = (n − 1)!.