Combinatorics

Combinatorics is a branch of mathematics that deals with finite or countably infinite discrete structures and is therefore also attributed to the preamble discrete mathematics. Examples are graphs (graph theory ), some minor amounts such as associations, matroids, combinatorial designs, Latin squares, tessellations, permutations of objects, partitions. Differentiation from other fields of discrete mathematics is fluid. A definition of George Pólya referred to as the study of the combinatorics counting out, the existence and construction of configurations.

Depending on the methods and objects used distinction is also sub-disciplines such as algebraic combinatorics, analytic combinatorics, geometric and topological combinatorics, probabilistic combinatorics, combinatorial game theory, Ramsey theory. Especially with the optimization of discrete structures is concerned, the combinatorial optimization.

Historically, the combinatorics arose from Abzählproblemen of discrete structures such as the 17th century in the probability analysis of gambling occurred ( Blaise Pascal and others). This classic area of combinatorics is collectively referred to as enumerative combinatorics. Characteristic of the problems occurring in the enumerative combinatorics was that most new methods had to be devised for each single issue ad hoc. For a long time played the combinatorics therefore an outsider role in mathematics, summarizing theories of their sub-areas emerged only in the 20th century, for example in the schools of Gian- Carlo Rota and Richard P. Stanley.

The combinatorics has numerous applications in other areas of mathematics such as geometry, probability theory, algebra, set theory and topology in the computer science (for example, coding theory ) and theoretical physics, in particular in statistical mechanics.