Number of Triangulations of a M\"obius Strip

TL;DR

This paper derives a formula for counting all possible triangulations of a M"obius strip with n points on its edge and links this to the number of clusters in a related algebraic structure.

Contribution

It provides a closed-form formula for the number of triangulations of a M"obius strip with n points and connects this to quasi-cluster algebra clusters.

Findings

Number of triangulations given by 4^{n-1} + binom{2n-2}{n-1}

Established a connection with quasi-cluster algebra clusters

Enhanced understanding of combinatorial structures on non-orientable surfaces

Abstract

Consider a M\"obius strip with $n$ chosen points on its edge. A triangulation is a maximal collection of arcs among these points and cuts the strip into triangles. In this paper, we proved the number of all triangulations that one can obtain from a M\"obius strip with $n$ chosen points on its edge is given by $4^{n-1}+\binom{2n-2}{n-1}$, then we made the connection with the number of clusters in the quasi-cluster algebra arising from the M\"obius strip.