Abstract :
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.
