Interface line operators in $\mathcal{N}=4$ SYM theories and supersymmetric indices

We study configurations of two $\mathcal{N}=4$ super Yang-Mills theories of unitary gauge groups connected by the BPS interfaces involving line operators. We find strong evidence of S-duality of the configurations as precise matching of the line defect half-indices which enumerate the BPS local operators at the junctions of the interfaces and line operators. The interface line defect half-indices are expressible as intriguing closed-form formulae involving the $q$-binomial coefficients and the principal specializations of the Schur functions.