Asymptotic enumeration and logical limit laws for expansive multisets and selections

Given a sequence of integers aj, j ≥ 1, a multiset is a combinatorial object composed of unordered components, such that there are exactly aj one-component multisets of size j. When aj ≍ jr−1yj for some r > 0, y ≥ 1, then the multiset is called expansive. Let cn be the number of multisets of total size n. Using a probabilistic approach, we prove for expansive multisets that cn/cn+1 → 1 and that cn/cn+1 < 1 for large enough n. This allows us to prove Monadic Second Order Limit Laws for expansive multisets. The above results are extended to a class of expansive multisets with oscillation. Moreover, under the condition aj = Kj r−1yj+O(yνj), where K > 0, r > 0, y > 1, ν ∈ (0, 1), we find an explicit asymptotic formula for cn. In a similar way we study the asymptotic behavior of selections which are defined as multisets composed of components of distinct sizes. Department of Mathematics, Technion-Israel Institute of Technology, Haifa, 32000, Israel, e-mail:[email protected] School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, United Kingdom, e-mail:[email protected] 2000 Mathematics Subject Classification: 6 0C05 (primary), 05A16 (secondary) 1 1 Summary and Historical remarks Given a sequence of integers aj ≥ 0, j ≥ 1, a multiset is a combinatorial object of finite total size composed of unordered indecomposable components such that there are exactly aj single component multisets of size j. There is no restriction on the number of times a component may appear in the multiset.In view of this, let