论文标题
关于第二种独立收集过程的至少独立收集过程
On the minimum of independent collecting processes via the Stirling numbers of the second kind
论文作者
论文摘要
我们考虑了组合问题，其中$ p $播放器的目标是统一分布的一组$ n $不同类型的物品（物种）。让随机变量$ t_ {n（i）}，\，\，\，i = 1,2，\ cdots，p $表示所需的试验次数，直到每个播放器分别检测到所有$ n $类型（至少一次）。本文研究了数字$ p $在期望的渐近学，第二刻的渐近学中的影响，以及随机变量\ begin \ begin {equation*} m_ {n（p）}：= \ bigwedge_ {i = 1}^p t_} \ end {equation*}这些量化表达式的主要成分是涉及第二类的stirl数字的总和；为此探讨了渐近学。在论文的最后，我们在交替的二项式总和上猜想了出色的\ textit {combinatorial Identity}。这些总和已由P. Flajolet（主要是）研究，因为它们应用于数字搜索树和Quadtrees。
We consider the combinatorial problem where $p$ players aim to a complete set of $N$ different types of items (species) which are uniformly distributed. Let the random variables $T_{N(i)},\,\,i=1,2,\cdots,p$ denoting the number of trials needed until all $N$ types are detected (at least once), respectively for each player. This paper studies the impact of the number $p$ in the asymptotics of the expectation, the second moment, and the variance of the random variable \begin{equation*} M_{N(p)}: = \bigwedge_{i=1}^p T_{N(i)},\,\,\,\,\,\,N\rightarrow \infty. \end{equation*} The main ingredient in the expression of these quantittes are sums involving the Stirling numbers of the second kind; for which the asymptotics are explored. At the end of the paper we conjecture on a remarkable \textit{combinatorial identity}, regarding alternating binomial sums. These sums have been studied (mainly) by P. Flajolet due to their applications to digital search trees and quadtrees.
