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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.
[7] [8] The number R of complexions is = (+)!! ()!. The graphical representation of each possible distribution would contain P copies of the symbol ε and N – 1 copies of the symbol 0. In their demonstration, Ehrenfest and Kamerlingh Onnes took N = 4 and P = 7 (i.e., R = 120 combinations
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.
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.
The rook polynomial R B (x) of a board B is the generating function for the numbers of arrangements of non-attacking rooks: = = (,) (),where () is the number of ways to place k non-attacking rooks on the board B.
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.
The discipline of combinatorial topology used combinatorial concepts in topology and in the early 20th century this turned into the field of algebraic topology.. In 1978 the situation was reversed—methods from algebraic topology were used to solve a problem in combinatorics—when László Lovász proved the Kneser conjecture, thus beginning the new field of topological combinatorics.