Algebraic combinatorics

[1] Through the early or mid-1990s, typical combinatorial objects of interest in algebraic combinatorics either admitted a lot of symmetries (association schemes, strongly regular graphs, posets with a group action) or possessed a rich algebraic structure, frequently of representation theoretic origin (symmetric functions, Young tableaux).

This period is reflected in the area 05E, Algebraic combinatorics, of the AMS Mathematics Subject Classification, introduced in 1991.

Thus the combinatorial topics may be enumerative in nature or involve matroids, polytopes, partially ordered sets, or finite geometries.

Among other things, this ring plays an important role in the representation theory of the symmetric groups.

Association schemes provide a unified approach to many topics, for example combinatorial designs and coding theory.

Their theory was further developed by many mathematicians, including Percy MacMahon, W. V. D. Hodge, G. de B. Robinson, Gian-Carlo Rota, Alain Lascoux, Marcel-Paul Schützenberger and Richard P. Stanley.

A matroid is a structure that captures and generalizes the notion of linear independence in vector spaces.