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Given two nonlinear input-output systems which are analytic in the sense that they have convergent Fliess operator representations, it is known that their interconnection in almost any fashion will produce another system in this same class. Recent work has focused on characterizing the radius of convergence for a variety of such interconnections. A key observation in this analysis was that certain combinatorial integer sequences naturally appear. The first goal of this paper is to gather from the literature all the known relationships between system interconnections and such sequences and organize them in a coherent manner. In the process it becomes clear that Stirling numbers play a central role in the most nontrivial types of system interconnections, namely, cascade and feedback connections. The second goal is to describe these relationships.