Evaluating the generalized Buchshtab function and revisiting the variance of the distribution of the smallest components of combinatorial objects

TL;DR

This paper investigates the distribution of the smallest components in various combinatorial objects, revisiting variance calculations and exploring the generalized Buchstab function for different parameters using multiple analytical and computational methods.

Contribution

It provides new methods for evaluating the generalized Buchstab function and refines the understanding of the variance of smallest components in combinatorial objects.

Findings

Revised estimate of the variance coefficient for the smallest component size.

Development of new analytical and numerical techniques for evaluating the Buchstab function.

Exact distribution computations for objects up to size 4000.

Abstract

Let $n\geq 1$ and $X_{n}$ be the random variable representing the size of the smallest component of a random combinatorial object made of $n$ elements. A combinatorial object could be a permutation, a monic polynomial over a finite field, a surjective map, a graph, and so on. By a random combinatorial object, we mean a combinatorial object that is chosen uniformly at random among all possible combinatorial objects of size $n$. It is understood that a component of a permutation is a cycle, an irreducible factor for a monic polynomial, a connected component for a graph, etc. Combinatorial objects are categorized into parametric classes. In this article, we focus on the exp-log class with parameter $K=1$ (permutations, derangements, polynomials over finite field, etc.) and $K=1/2$ (surjective maps, $2$-regular graphs, etc.) The generalized Buchstab function $\Omega_{K}$ plays an important role in evaluating probabilistic and statistical quantities. For $K=1$, Theorem $5$ from \cite{PanRic_2001_small_explog} stipulates that $\mathrm{Var}(X_{n})=C(n+O(n^{-\epsilon}))$ for some $\epsilon>0$ and sufficiently large $n$. We revisit the evaluation of $C=1.3070\ldots$ using different methods: analytic estimation using tools from complex analysis, numerical integration using Taylor expansions, and computation of the exact distributions for $n\leq 4000$ using the recursive nature of the counting problem. In general for any $K$, Theorem $1.1$ from \cite{BenMasPanRic_2003} connects the quantity $1/\Omega_{K}(x)$ for $x\geq 1$ with the asymptotic proportion of $n$-objects with large smallest components. We show how the coefficients of the Taylor expansion of $\Omega_{K}(x)$ for $\lfloor x\rfloor \leq x < \lfloor x\rfloor+1$ depends on those for $\lfloor x\rfloor-1 \leq x-1 < \lfloor x\rfloor$. We use this family of coefficients to evaluate $\Omega_{K}(x)$.