  Gibbs partition models are the largest class of infinite exchangeable
partitions of the positive integers generalizing the product form of the
probability function of the two-parameter Poisson-Dirichlet family. Recently
those models have been investigated in a Bayesian nonparametric approach to
species sampling problems as alternatives to the Dirichlet and the Pitman-Yor
process priors. Here we derive marginals of conditional and unconditional
multivariate distributions arising from exchangeable Gibbs partitions to obtain
explicit formulas for joint falling factorial moments of corresponding
conditional and unconditional Gibbs sampling formulas. Our proofs rely on a
known result on factorial moments of sum of non independent indicators. We
provide an application to a Bayesian nonparametric estimation of the predictive
probability to observe a species already observed a certain number of times.
