Two Combinatorial Applications of the Aleksandrov-Fenchel Inequalities

The Aleksandrov-Fenchel inequalities from the theory of mixed volumes are used to prove that certain sequences of combinatorial interest are log concave (and therefore unimodal). 1. MIXED VOLUMES We wish to show how the Aleksandrov-Fenchel inequalities from the theory of mixed volumes can be used to prove that certain sequences of combinatorial interest are log concave (and therefore unimodal). In particular, we prove the following two results (all terminology will be defined later): (a) Let M be a unimodular (= regular) matroid of rank n on a finite set S, and let T G S. Let fi be the number of bases B of M satisfying concave. (b) Let P be a finite poset (= partially ordered set) with n elements, and let x E P. Let Ni be the number of order-preserving bijections We first review the salient facts from the theory of mixed volumes.