Faà di Bruno’s formula and the distributions of random partitions in population genetics and physics

We show that the formula of Faà di Bruno for the derivative of a composite function gives, in special cases, the sampling distributions in population genetics that are due to Ewens and to Pitman. The composite function is the same in each case. Other sampling distributions also arise in this way, such as those arising from Dirichlet, multivariate hypergeometric, and multinomial models, special cases of which correspond to Bose–Einstein, Fermi–Dirac, and Maxwell–Boltzmann distributions in physics. Connections are made to compound sampling models.