  We examine "partition zeta functions" analogous to the Riemann zeta function
but summed over subsets of integer partitions. We prove an explicit formula for
a family of partition zeta functions already shown to have nice properties --
those summed over partitions of fixed length -- which yields complete
information about analytic continuation, poles and trivial roots of the zeta
functions in the family. Then we present a combinatorial proof of the explicit
formula, which shows it to be a zeta function analog of MacMahon's partial
fraction decomposition of the generating function for partitions of fixed
length.
