Enumerating triangulations by parallel diagonals

TL;DR

This paper develops a method to enumerate triangulations of convex polygons based on the count of diagonals parallel to a fixed edge, utilizing the Shapiro convolution identity.

Contribution

It introduces a novel enumeration approach for triangulations using the Shapiro convolution identity and provides an interpretation of this identity in the context of triangulations.

Findings

Derived explicit formulas for enumeration based on parallel diagonals.

Connected triangulation enumeration to the Shapiro convolution identity.

Provided combinatorial interpretations of algebraic identities.

Abstract

The triangulations of a regular convex polygon are enumerated according to the number of diagonals parallel to a fixed edge. The enumeration uses the Shapiro convolution identity, as well as an interpretation of this identity in terms of triangulations.