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

Claude GravelDaniel Panario
Dec 2022
Let $n\geq 1$ and $X_{n}$ be the random variable representing the size of thesmallest component of a combinatorial object generated uniformly and randomlyover $n$ elements. A combinatorial object could be a permutation, a monicpolynomial over a finite field, a surjective map, a graph, and so on. It isunderstood that a component of a permutation is a cycle, an irreducible factorfor a monic polynomial, a connected component for a graph, etc. Combinatorialobjects are categorized into parametric classes. In this article, we focus onthe exp-log class with parameter $K=1$ (permutations, derangements, polynomialsover finite field, etc.) and $K=1/2$ (surjective maps, $2$-regular graphs,etc.) The generalized Buchshtab function $\Omega_{K}$ plays an important rolein 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 $\epsilon>0$ 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$\Omega_{K}(x)$.