  We propose a set of novel expansions of Nekrasov's instanton partition
functions. Focusing on 5d supersymmetric pure Yang-Mills theory with unitary
gauge group on $\mathbb{C}^2_{q,t^{-1}} \times \mathbb{S}^1$, we show that the
instanton partition function admits expansions in terms of partition functions
of unitary gauge theories living on the 3d subspaces $\mathbb{C}_{q} \times
\mathbb{S}^1$, $\mathbb{C}_{t^{-1}} \times \mathbb{S}^1$ and their intersection
along $\mathbb{S}^1$. These new expansions are natural from the BPS/CFT
viewpoint, as they can be matched with $W_{q,t}$ correlators involving an
arbitrary number of screening charges of two kinds. Our constructions
generalize and interpolate existing results in the literature.
