  We compute the partition functions of $\mathcal{N} = 1$ gauge theories on
$S^2 \times \mathbb{R}^2_\varepsilon$ using supersymmetric localization. The
path integral reduces to a sum over vortices at the poles of $S^2$ and at the
origin of $\mathbb{R}^2_\varepsilon$. The exact partition functions allow us to
test Seiberg duality beyond the supersymmetric index. We propose the
$\mathcal{N} = 1$ partition functions on the $\Omega$-background, and show that
the Nekrasov partition functions can be recovered from these building blocks.
