  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
$\Omega_{K}(x)$.
