The expected number of alleles in a gene conversion model with mutation

Gene conversions and mutations affect the number of different alleles in a population as it evolves. Here these effects are studied for a sample of r individuals represented by one chromosome, each with n loci. The long-term evolution of the sample is modelled with the coalescent. Gene conversions can only occur within a chromosome between a pair of loci. Recombination is not included in the model. Mutations occur according to the infinitely many alleles model. A new expression for the expected number of alleles is derived using the Backward time approach. It is then possible to simulate the expectation for various sample paths. The new estimator is computationally inexpensive and has a lower variance than the conventional, forward time, approximation by simulation.