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Given a surface with boundary and some points on its boundary, a polygondiagram is a way to connect those points as vertices of non-overlappingpolygons on the surface. Such polygon diagrams represent non-crossingpermutations on a surface with any genus and number of boundary components. Ifonly bigons are allowed, then it becomes an arc diagram. The count of arcdiagrams is known to have a rich structure. We show that the count of polygondiagrams exhibits the same interesting behaviours, in particular it is almostpolynomial in the number of points on the boundary components, and the leadingcoefficients of those polynomials are the intersection numbers on thecompactified moduli space of curves.

Counting non-crossing permutations on surfaces of any genus