  We introduce and study multiple partition structures which are sequences of
probability measures on families of Young diagrams subjected to a consistency
condition. The multiple partition structures are generalizations of Kingman's
partition structures, and are motivated by a problem of population genetics.
They are related to harmonic functions and coherent systems of probability
measures on a certain branching graph. The vertices of this graph are multiple
Young diagrams (or multiple partitions), and the edges depend on the Jack
parameter. Our main result establishes a bijective correspondence between the
set of harmonic functions on the graph and probability measures on the
generalized Thoma set. The correspondence is determined by a canonical integral
representation of harmonic functions. As a consequence we obtain a
representation theorem for multiple partition structures.
  We give an example of a multiple partition structure which is expected to be
relevant for a model of population genetics for the genetic variation of a
sample of gametes from a large population. Namely, we construct a probability
measure on the wreath product of a finite group with the symmetric group. The
constructed probability measure defines a multiple partition structure which is
a generalization of the Ewens partition structure studied by Kingman. We show
that this multiple partition structure can be represented in terms of a
multiple analogue of the Poisson-Dirichlet distribution called the multiple
Poisson-Dirichlet distribution in the paper.
