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 = Kjr−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 combinatorial objects composed of unordered components of distinct sizes.