On the number of representations of matroids over finite fields (Q679433)

For a finite field \(F_q\), \(n_q(M)\) denotes the number of isomorphism classes of representations of a matroid \(M\) over \(F_q\). Relations between the \(n_q(M)\) for various \(M\) are explored, using ideas of Glynn who studied the numbers of configurations of points in projective planes [see \textit{D. G. Glynn}, Rings of geometries. II, J. Comb. Theory, Ser. A 49, No. 1, 26-66 (1988; Zbl 0661.51009)]. These relations generate an induction process which gives an expression of \(n_q(M)\) in terms of the \(n_q(M_i)\), where the \(M_i\) are the matroids of smaller rank and corank, which cannot be further simplified. The expression is linear in the \(n_q(M_i)\) with integer polynomials in \(q\) as coefficients. In the last section, the author reproduces some formulas of \(n_q(U_{k,n})\), for particular \(k\) and \(n\), where \(U_{k,n}\) is the uniform matroid used for MDS-codes of length \(n\) and dimension \(k\).