Asymptotic growth of the number of curves with bounded length on hyperbolic surfaces has been thoroughly studied and is still an active area of research. Starting from Huber’s Prime Geodesic Theorem, to Mirzakhani’s counting of closed curves of a given type, arriving at Erlandsson-Souto’s generalization to homogeneous functions on geodesic currents, passing through other work with e.g. Parlier and McShane-Rivin among many others. We study the combinatorial version of the problem, changing hyperbolic length for word-length. This will still have consequences on hyperbolic surfaces and will allow us to not just study the asymptotics, but the exact counting for any given length. We classify closed curves on a once-punctured torus with a single self-intersection from a combinatorial perspective. We determine the number of closed curves with given word-length and with zero, one, and arbitrary self-intersections.
