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.