Counting Triangulations of Almost-Convex Polygons

We define an almost-convex polygon as a non-convex polygon in which any two vertices see each other inside the polygon unless they are not adjacent and belong to a chain of consecutive concave vertices. Using inclusion-exclusion techniques, we find formulas for the number of triangulations of almost-convex polygons in terms of the number and position of the concave vertices. We translate these formulas into the language of generating functions and provide several simple asymptotic estimates. We also prove that certain balanced configurations yield the maximum number of triangulations.