On the minimum of independent collecting processes via the Stirling numbers of the second kind

TL;DR

This paper investigates the asymptotic behavior of the minimum number of trials needed by multiple players to collect all item types, using Stirling numbers of the second kind, and explores related combinatorial identities.

Contribution

It provides new asymptotic results for the expectation, variance, and second moment of the minimum collection time involving Stirling numbers, and conjectures a novel combinatorial identity.

Findings

Asymptotic formulas for the minimum collection time are derived.

The impact of the number of players on collection statistics is characterized.

A conjecture on a new combinatorial identity related to Stirling numbers is proposed.

Abstract

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.