  We study a family of inhomogeneous Ising chain models along with an
equivalent family of nearest neighbour particle systems. By the correspondence
between the two families we prove identities of combinatorial significance
relating to certain integer and Frobenius partitions. In particular, for
certain parameter values we see that one of our identities relates to
generating functions for overpartitions. Using the identities we give a
surprising product form of the partition function for an Ising chain with
homogeneous interaction and an inhomogeneous external field. We also use the
connection between the Ising chain and particle system to find interesting
long-range reversible dynamics for the particle system that do not have a
product form stationary measure.
