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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, the twelvefold way is a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical problems of counting permutations, combinations, multisets, and partitions either of a set or of a number.
Proofs That Really Count: the Art of Combinatorial Proof is an undergraduate-level mathematics book on combinatorial proofs of mathematical identies.That is, it concerns equations between two integer-valued formulas, shown to be equal either by showing that both sides of the equation count the same type of mathematical objects, or by finding a one-to-one correspondence between the different ...
Lookahead is an important component of combinatorial search, which specifies, roughly, how deeply the graph representing the problem is explored. The need for a specific limit on lookahead comes from the large problem graphs in many applications, such as computer chess and computer Go.
A minimum spanning tree of a weighted planar graph.Finding a minimum spanning tree is a common problem involving combinatorial optimization. Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, [1] where the set of feasible solutions is discrete or can be reduced to a discrete set.
Combinatorics has always played an important role in quantum field theory and statistical physics. [3] However, combinatorial physics only emerged as a specific field after a seminal work by Alain Connes and Dirk Kreimer , [ 4 ] showing that the renormalization of Feynman diagrams can be described by a Hopf algebra .
Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations .
Combinatorics, a MathWorld article with many references. Combinatorics, from a MathPages.com portal. The Hyperbook of Combinatorics, a collection of math articles links. The Two Cultures of Mathematics by W. T. Gowers, article on problem solving vs theory building