论文标题
评估广义buchshtab函数并重新审视组合物体最小组件的分布的方差
Evaluating the generalized Buchshtab function and revisiting the variance of the distribution of the smallest components of combinatorial objects
论文作者
论文摘要
令$ n \ geq 1 $和$ x_ {n} $为随机变量，代表由$ n $元素制成的随机组合对象的最小组件的大小。组合对象可以是一个置换，是有限场上的一元多项式，汇总图，图等。通过随机组合对象，我们的意思是一个组合对象，该对象在大小$ n $的所有可能组合对象中均匀地选择。可以理解的是，排列的一个组件是一个循环，是一元多项式，图形的连接组件等。组合对象分类为参数类别。在本文中，我们专注于带有参数$ k = 1 $的Exp-log类（排列，扰动，对有限字段的多项式等）和$ k = 1/2 $（溢流图，$ 2 $ - 台式图等）广义buchstab功能$ω__{k} $在评估概率和宣传量中起重要作用。对于$ k = 1 $，定理$ 5 $来自\ cite {panric_2001_small_explog}根据某些$ε> 0 $和足够的大$ n $。我们使用不同的方法对$ C = 1.3070 \ ldots $进行评估：使用复杂分析中的工具，使用Taylor扩展的工具进行分析估算，以及使用计数问题的递归性质的$ N \ LEQ 4000 $计算精确分布的计算。通常，对于任何$ k $，定理$ 1.1 $从\ cite {benmaspanric_2003}连接数量$ 1/ω_{k}（x）$ for $ x \ geq 1 $，渐近比例为$ n $ - 对象的比例为$ n $ -Objects，该比例为$ n $ objects。我们展示了taylor扩展的系数$ω_{k}（x）$，for $ \ lfloor x \ rfloor \ leq x <\ lfloor x \ rfloor+1 $取决于$ \ lfloor x \ rfloor x \ rfloor x \ rfloor-1 \ leq x-1 <\ leq x-1 <\ lfloor x \ rfloor x \ rfloor x \ rfloor x我们使用这个系数系列来评估$ω_{k}（x）$。
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 $Ω_{K}$ plays an important role in evaluating probabilistic and statistical quantities. For $K=1$, Theorem $5$ from \cite{PanRic_2001_small_explog} stipulates that $\mathrm{Var}(X_{n})=C(n+O(n^{-ε}))$ for some $ε>0$ 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 $1/Ω_{K}(x)$ 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 $Ω_{K}(x)$ 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 $Ω_{K}(x)$.
