Combinatorics of Vector Spaces over Finite Fields

Abstract

In this writing we examine some known results concerning exact enumeration of vector spaces whose underlying fields are finite. We explore some of the many interesting analogies between subsets and subspaces. We derive some q-analogue identities and theorems. We link these results to some previously existing, as well as some newly added, sequences in the On Line Encyclopedia of Integer Sequences, OEIS. We give generating functions, formulas, recurrences, and Mathematica code that count various statistics concerning vector spaces over finite fields, linear opera-tors, and subspace lattices.