  A formula was recently proposed for the perturbative partition function of
certain $\mathcal N=1$ gauge theories on the round four-sphere, using an
analytic-continuation argument in the number of dimensions. These partition
functions are not currently accessible via the usual
supersymmetric-localisation technique. We provide a natural refinement of this
result to the case of the ellipsoid. We then use it to write down the
perturbative partition function of an $\mathcal N=1$ toroidal-quiver theory (a
double orbifold of $\mathcal N=4$ super Yang-Mills) and show that, in the
deconstruction limit, it reproduces the zero-winding contributions to the BPS
partition function of (1,1) Little String Theory wrapping an emergent torus. We
therefore successfully test both the expressions for the $\mathcal N=1$
partition functions, as well as the relationship between the toroidal-quiver
theory and Little String Theory through dimensional deconstruction.
