  We study a shift action defined on multipartitions and on residue multisets
of their Young diagrams. We prove that the minimal orbit cardinality among all
multipartitions associated to a given multiset depends only on the orbit
cardinality of the multiset. Using abaci, this problem reduces to a convex
optimisation problem over the integers with linear constraints. We solve it by
proving an existence theorem for binary matrices with prescribed row, column
and block sums. Finally, we give some applications to the representation theory
of the Hecke algebra of the complex reflection group $G(r,p,n)$.
