On coefficients of the Tutte polynomial

This paper characterizes, for each i and j, the matroids that are minor-minimal among connected matroids M with b(ij)(M) > 0, where t(M) = Sigma b(ij)(M)x(i)y(j) is the Tutte polynomial of M. One consequence of this characterization for a connected matroid M is that b(11)(M) > 0 if and only if the two-wheel is a minor of M. Similar results are obtained for other small values of i and j. A generalization of these results leads to new combinatorial proofs which strengthen known results on the coefficients. These results imply that if M is simple and representable over GF(q), then there are coefficients of its Tutte polynomial which count the flats of M of each rank that are projective spaces. Similarly, for a simple graphic matroid M(G), there are coefficients that count the number of cliques of each size contained in G. (C) 1998 Elsevier Science B.V. All rights reserved.