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An asymptotic sampling formula for the coalescent with recombination. (English) Zbl 1193.92077
Summary: The W. J. Ewens sampling formula (ESF) [Theor. Popul. Biol. 3, 87–112 (1972; Zbl 0245.92009)] is a one-parameter family of probability distributions with a number of intriguing combinatorial connections. This elegant closed-form formula first arose in biology as the stationary probability distribution of a sample configuration at one locus under the infinite-alleles model of mutations. Since its discovery, the ESF has been used in various biological applications, and has sparked several interesting mathematical generalizations. In the population genetics community, extending the underlying random-mating model to include recombination has received much attention in the past, but no general closed-form sampling formula is currently known even for the simplest extension, that is, a model with two loci.
We show that it is possible to obtain useful closed-form results in the case the population-scaled recombination rate $$\rho$$ is large but not necessarily infinite. Specifically, we consider an asymptotic expansion of the two-locus sampling formula in inverse powers of $$\rho$$ and obtain closed-form expressions for the first few terms in the expansion. Our asymptotic sampling formula applies to arbitrary sample sizes and configurations.

##### MSC:
 92D15 Problems related to evolution 92D10 Genetics and epigenetics 60C05 Combinatorial probability 65C50 Other computational problems in probability (MSC2010)
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