  We show that the chaos representation of some Compound Poisson Type processes
displays an underlying intrinsic combinatorial structure, partly independent of
the chosen process. From the computational viewpoint, we solve the arising
combinatorial complexity by means of the moments/cumulants duality for the laws
of the corresponding processes, themselves measures on distributional spaces,
and provide a combinatorial interpretation of the associated 'extended' Fock
spaces. From the theoretical viewpoint, in the case of the Gamma measure, we
trace back such complexity to its 'simplicial part', i.e. the
Dirichlet-Ferguson measure, hence to the Dirichlet distribution on the
finite-dimensional simplex. We thoroughly explore the combinatorial and
algebraic properties of the latter distribution, arising in connection with
cycle index polynomials of symmetric groups and dynamical symmetry algebras of
confluent Lauricella functions.
