Moment Statistics in the Boltzmann Probability Model

Abstract

A rich family of labeled as well as unlabeled combinatorial structures is accessible by using so-called admissible specifications for which direct access to (counting) generating function equations exists. Using methods from analytic combinatorics, this quite often provides access to asymptotics for their coefficients and thus average-case statistics and knowledge on higher moments and distributions of structural parameters. Furthermore, admissible specifications are the foundation for different approaches of random sampling algorithms either uniformly for a fixed size (e.g., by unranking a random rank) or for random sizes in the Boltzmann model. The latter is of special interest for its efficiency in case of approximate size sampling. It is standard to derive asymptotics for moments from coefficients of generating functions for analytical purposes, e.g. using the saddle-point or the 𝒪-transfer method. In this paper we highlight connections between such asymptotics and the values of generating functions as computed for Boltzmann samplers. We show their use to derive fixed-size statistics from random-size Boltzmann samples. Furthermore, we introduce a new approach for the (leading term) average-case (and higher moment) analysis of structural parameters of combinatorial objects that makes the computation of generating function coefficients superfluous.