An Introduction to $q$-Species

Abstract

The combinatorial theory of species developed by Joyal provides a foundation for enumerative combinatorics of objects constructed from finite sets. In this paper we develop an analogous theory for the enumerative combinatorics of objects constructed from vector spaces over finite fields. Examples of these objects include subspaces, flags of subspaces, direct sum decompositions, and linear maps or matrices of various types. The unifying concept is that of a "$q$-species", defined to be a functor from the category of finite dimensional vector spaces over a finite field to the category of finite sets.