A multiindexed sturm sequence of polynomials and unimodality of certain combinatorial sequences

With any multiset n we associate the numbers $\text{O}$(n, k) of compositions of n into exactly k parts. The polynomials k_n(x) = Σ_kO(n, k)x^k are shown to form a multiindexed Sturm sequence over (?1, 0). As consequences we obtain the unimodality of the sequence {O(n, k)}_k for any n, of the generalized Eulerian numbers, and of the number of compositions of n with certain supplementary conditions imposed on the parts. The strong logarithmic concavity of the Stirling numbers of the second kind also follows as a corollary.