On the genus filtration of diagrams over two backbones

In this paper we compute the bivariate generating function of $\gamma$-matchings over two backbones, filtered by the number of arcs and the topological genus. $\gamma$-matchings over two backbones are chord-diagrams, obtained via concatenation and nesting of irreducible shapes of topological genus $\le \gamma$. We show that the key information is contained in the polynomials counting these shapes and provide recursions that allow to compute the latter. In particular we give a bijection between such irreducible shapes over one and two backbones. We present two applications of our results. The first is concerned with RNA-RNA interaction structures, obtained from the $\gamma$-matchings via symbolic methods. We secondly show that, using analytic-combinatorial methods, the topological genus satisfies a central limit theorem.