Mathematical Sciences: Random Access Approximations in Combinatorics and Number Theory

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9626412 Arratia/Tavare ABSTRACT The investigators study the probabilistic properties of some of the most basic structures in combinatorics and number theory, such as partitions of sets and integers, and the factorization of uniformly chosen integers into primes. They apply techniques learned from the study of genetics and random permutations, including couplings, the total variation distance and Wasserstein metric for processes observing some or all of the possible component sizes, size-biased permutations of the components, sampling properties for populations, and Stein's method. The interaction between pure and applied method can be fruitful for both. Techniques motivated by the needs and intuitions from each field can illuminate the other. For example, the applied study of genetics naturally involves sampling from populations, which leads to the notion of a size- biased permutation, which simplifies the understanding of a question from pure math, involving the large prime factors of a random integer. In the other direction, results proved about primes, such as bounds on the "total variation distance", anticipated by three decades important work in statistics on the comparison of sampling with and without replacement. The proposed investigation attempts to straddle the boundary between pure and applied math as it relates to genetics, sampling, and prime numbers.