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$$h$$-vectors of matroids and logarithmic concavity. (English) Zbl 1304.05013
Summary: Let $$M$$ be a matroid on $$E$$, representable over a field of characteristic zero. We show that $$h$$-vectors of the following simplicial complexes are log-concave: (1) The matroid complex of independent subsets of $$E$$. (2) The broken circuit complex of $$M$$ relative to an ordering of $$E$$.
The first implies a conjecture of Colbourn on the reliability polynomial of a graph, and the second implies a conjecture of Hoggar on the chromatic polynomial of a graph. The proof is based on the geometric formula for the characteristic polynomial of Denham, Garrousian, and Schulze.

##### MSC:
 05B35 Combinatorial aspects of matroids and geometric lattices 52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
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