Abstract
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) = Σb 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.
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