  We derive exact formulae for the allele frequency spectrum under the
coalescent with mutation, conditioned on allele counts at some fixed time in
the past. We consider unlinked biallelic markers mutating according to a finite
sites, or infinite sites, model. This work extends the coalescent theory of
unlinked biallelic markers, enabling fast computations of allele frequency
spectra in multiple populations. Our results have applications to demographic
inference, species tree inference, and the analysis of genetic variation in
closely related species more generally.
