  We introduce a family of multivariate continuous-time pure birth and pure
death chains, with birth and death rates defined in terms of the generalized
binomial coefficients for multiplicity free actions. The state spaces for some
of the introduced processes are some sets of partitions (equivalently, Young
diagrams). The chains turn out to be the classical Markov processes obtained by
restricting Biane's quantum Ornstein-Uhlenbeck semigroups to commutative
C*-algebras related to Gelfand pairs built on Heisenberg groups.
