combinatorische

Combinatorische is the German adjective meaning "combinatorial" and is used to describe problems, methods, and theories in the mathematical field of combinatorics. It concerns discrete structures—such as sets, sequences, graphs, and codes—and the ways these objects can be counted, arranged, selected, or optimized. In German-language mathematics, phrases like kombinatorische Optimierung (combinatorial optimization) and kombinatorische Zahlentheorie appear.

Key topics include permutations, combinations, and binomial coefficients; graph theory; design theory; posets; coding theory; and block designs. Enumerative combinatorics counts objects; extremal combinatorics studies maximal or minimal configurations; algebraic and geometric combinatorics connect combinatorics to algebra and geometry; probabilistic combinatorics uses randomness to prove existence or estimate size; and combinatorial optimization seeks efficient constructions and algorithms.

Common methods include bijective proofs, generating functions, recurrence relations, inclusion–exclusion, and the probabilistic method; Polya's enumeration theorem and Burnside's lemma are used to count configurations up to symmetry.

Historically, the field has roots in early counting problems and permutation theory; Euler's work on permutations and the binomial theorem laid foundations, with further development in the 20th century by Polya and Erdos.

Applications span computer science, coding theory, cryptography, scheduling, network design, and experimental design in the sciences; combinatorial methods also influence chemistry and economics.

As a core part of discrete mathematics, combinatorics intersects with number theory, algebra, probability, and optimization, and underpins many algorithms and data-structure analyses.