Expansive Multisets: Asymptotic Enumeration

TL;DR

This paper derives asymptotic formulas for counting multisets of combinatorial objects with a focus on phase transitions depending on the ratio of total size to number of components, using probabilistic and analytic methods.

Contribution

It introduces a novel combination of probabilistic and analytic techniques to analyze the asymptotic enumeration of multisets with a phase transition phenomenon.

Findings

Identifies a phase transition in the structure of the counting formula.

Provides asymptotic enumeration formulas for all ranges of N.

Demonstrates a passage from a bivariate local limit to a univariate limit.

Abstract

Consider a non-negative sequence $c_n = h(n) \cdot n^{\alpha-1} \cdot \rho^{-n}$, where $h$ is slowly varying, $\alpha>0$, $0<\rho<1$ and $n\in\mathbb{N}$. We investigate the coefficients of $G(x,y) = \prod_{k\ge1}(1-x^ky)^{-c_k}$, which is the bivariate generating series of the multiset construction of combinatorial objects. By a powerful blend of probabilistic methods based on the Boltzmann model and analytic techniques exploiting the well-known saddle-point method we determine the number of multisets of total size $n$ with $N$ components, that is, the coefficient of $x^ny^N$ in $G(x,y)$, asymptotically as $n\to\infty$ and for all ranges of $N$. Our results reveal a phase transition in the structure of the counting formula that depends on the ratio $n/N$ and that demonstrates a prototypical passage from a bivariate local limit to an univariate one.