  We study 7D maximally supersymmetric Yang-Mills theory on 3-Sasakian
manifolds. For manifolds whose hyper-K\"ahler cones are hypertoric we derive
the perturbative part of the partition function. The answer involves a special
function that counts integer lattice points in a rational convex polyhedral
cone determined by hypertoric data. This also gives a more geometric structure
to previous enumeration results of holomorphic functions in the literature.
Based on physics intuition, we provide a factorisation result for such
functions. The full proof of this factorisation using index calculations will
be detailed in a forthcoming paper.
