Factorization of the 3d superconformal index with an adjoint matter

We work out the factorization of the 3d superconformal index for N=2 U(N c ) gauge theory with one adjoint chiral multiplet as well as N f fundamental, N a anti-fundamental chiral multiplets. Using the factorization, one can prove the Seiberg-like duality for N=4 U(N c ) theory with N f hypermultiplets at the index level. We explicitly show that monopole operators violating unitarity bound in a bad theory are mapped to free hypermultiplets in the dual side. For N=2 U(N c ) theory with one adjoint matter X, N f fundamental, N a anti-fundamental chiral multiplets with superpotential W = trX n+1, we work out Seiberg-like duality for this theory. The index computation provides combinatorial identities for a dual pair, which we carry out intensive numerical checks.