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Let $n\geq 1$ and $X_{n}$ be the random variable representing the size of thesmallest component of a random combinatorial object made of $n$ elements. Acombinatorial object could be a permutation, a monic polynomial over a finitefield, 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 allpossible combinatorial objects of size $n$. It is understood that a componentof a permutation is a cycle, an irreducible factor for a monic polynomial, aconnected component for a graph, etc. Combinatorial objects are categorizedinto parametric classes. In this article, we focus on the exp-log class withparameter $K=1$ (permutations, derangements, polynomials over finite field,etc.) and $K=1/2$ (surjective maps, $2$-regular graphs, etc.) The generalizedBuchstab function $\Omega_{K}$ plays an important role in evaluatingprobabilistic 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 $\epsilon>0$ andsufficiently large $n$. We revisit the evaluation of $C=1.3070\ldots$ usingdifferent methods: analytic estimation using tools from complex analysis,numerical integration using Taylor expansions, and computation of the exactdistributions for $n\leq 4000$ using the recursive nature of the countingproblem. 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 asymptoticproportion of $n$-objects with large smallest components. We show how thecoefficients 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$\Omega_{K}(x)$.

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