Asymptotic Enumeration and Limit Laws for Multisets: the Subexponential Case

TL;DR

This paper analyzes the asymptotic enumeration and distribution of multisets constructed from combinatorial classes with subexponential growth, revealing a phenomenon called extreme condensation and providing uniform results for large parameters.

Contribution

It establishes asymptotic formulas for multisets with subexponential growth and uncovers the extreme condensation phenomenon in their component distribution.

Findings

Asymptotic size of multisets with fixed total and component counts

Discovery of the extreme condensation phenomenon

Uniform results for large total and component counts

Abstract

For a given combinatorial class $\mathcal{C}$ we study the class $\mathcal{G} = \mathrm{MSET}(\mathcal{C})$ satisfying the multiset construction, that is, any object in $\mathcal{G}$ is uniquely determined by a set of $\mathcal{C}$-objects paired with their multiplicities. For example, $\mathrm{MSET}(\mathbb{N})$ is (isomorphic to) the class of number partitions of positive integers, a prominent and well-studied case. The multiset construction appears naturally in the study of unlabelled objects, for example graphs or various structures related to number partitions. Our main result establishes the asymptotic size of the set $\mathcal{G}_{n,N}$ that contains all multisets in $\mathcal{G}$ having size $n$ and being comprised of $N$ objects from $\mathcal{C}$, as $n$ \emph{and} $N$ tend to infinity and when the counting sequence of $\mathcal{C}$ is governed by subexponential growth; this is a particularly important setting in combinatorial applications. Moreover, we study the component distribution of random objects from $\mathcal{G}_{n,N}$ and we discover a phenomenon that we baptise \emph{extreme condensation}: taking away the largest component as well as all the components of the smallest possible size, we are left with an object which converges in distribution as $n,N\to\infty$. The distribution of the limiting object is also retrieved. Moreover and rather surprisingly, in stark contrast to analogous results for labelled objects, the results here hold uniformly in $N$.