The perimeter generating function of punctured staircase polygons

TL;DR

This paper derives a long series expansion for the perimeter generating function of punctured staircase polygons using a transfer matrix approach, revealing it satisfies an order 8 linear Fuchsian differential equation.

Contribution

It introduces a transfer matrix method to analyze punctured staircase polygons and identifies the differential equation governing their generating function.

Findings

Generating function satisfies an order 8 Fuchsian differential equation

Series expansions can be reproduced from the differential equation

Analysis of the differential equation's properties was performed

Abstract

Using a simple transfer matrix approach we have derived very long series expansions for the perimeter generating function of punctured staircase polygons (staircase polygons with a single internal staircase hole). We find that all the terms in the generating function can be reproduced from a linear Fuchsian differential equation of order 8. We perform an analysis of the properties of the differential equation.