Bijective methods in the theory of finite vector spaces (Q794645)

The method, used to prove the following results is the observation that the canonical invariant of a vector subspace over a finite field is a matrix over the field, in reduced row echelon form (rref), whose rows span the subspace. To count subspaces is just count the matrices in rref. The results are: (a) a proof of a description, due to Pólya, of the coefficients of the Gaussian polynomials in terms of areas of certain lattice walks; (b) a proof of a three term recurrence relation satisfied by the ''Galois coefficients''; (c) an evaluation of the alternating sum of the Gaussian coefficients.