  We prove that 3d superconformal index for general $\mathcal N=2$ U(N) gauge
group with fundamentals and anti-fundmentals with/without Chern-Simons terms is
factorized into vortex and anti-vortex partition function. We show that for
simple cases, 3d vortex partition function coincides with a suitable
topological open string partition function. We provide much more elegant
derivation at the index level for $\mathcal N=2$ Seiberg-like dualities of
unitary gauge groups with fundamantal matters and $\mathcal N=4$ mirror
symmetry
