Symmetric Functions and Rectangular Catalan Combinatorics

Abstract

Symmetric functions provide one of the most efficient tools for combinatorial enumeration, in the context of objects that may be acted upon by permutations. Only assuming a basic knowledge of linear algebra, we introduce and describe the classical toolbox of symmetric functions, and give many examples of their uses. In particular, we present how classical formulas of enumerative combinatorics afford a natural generalization in terms of symmetric functions. We then rapidly present Macdonald symmetric functions and associated operators, to go on with the description of their relations to rectangular Catalan combinatorics. We also discuss the link with a realization of the elliptic Hall algebra, as an algebra of operators on symmetric functions.