ASYMPTOTIC ENUMERATION AND LOGICAL LIMIT LAWS FOR EXPANSIVE MULTISETS AND SELECTIONS

Abstract

Given a sequence of integers a_j, $j\ge 1$, a multiset is a combinatorial object composed of unordered components, such that there are exactly a_j one-component multisets of size j. When $a_j\asymp j^{r-1} y^j$ for some r > 0, $y\geq 1$, then the multiset is called expansive. Let c_n be the number of multisets of total size n. Using a probabilistic approach, we prove for expansive multisets that $c_n/c_{n+1}\to 1$ and that c_n/c_n+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 $a_j=Kj^{r-1}y^j + O(y^{\nu j})$, where K > 0, r > 0, y > 1, $\nu\in (0,1)$, we find an explicit asymptotic formula for c_n. In a similar way we study the asymptotic behavior of selections, which are defined as combinatorial objects composed of unordered components of distinct sizes.