On the probability that subset sequences are minimal

Let [n] denote {1,2,…,n} and let ω=(ω(i))pi=1 be a p-sequence of k-sets of [n]. If there is an element xi in ω(i) that is not in any other ω(j) with j≠i, then we call xi a representative of ω(i). If every entry of ω has t representatives, then we say that ω is (n,p,k,t)-minimal. In this paper, we give a lower bound for the probability that ω is (n,p,k,t)-minimal and we give the expected number of entries of ω with at least t representatives. These computations have practical applications. In some probabilistic group testing procedures, e.g., DNA library screening, they are used to analyze the likelihood of identifying the positives objects in a population.