Exact perimeter generating function for a model of punctured staircase polygons

TL;DR

This paper derives an exact, closed-form perimeter generating function for a model of punctured staircase polygons, combining series expansion, differential equations, and combinatorial proofs, with implications for generalized polygon models.

Contribution

It presents the first exact closed-form solution for the perimeter generating function of a punctured staircase polygon model, using novel differential equation and combinatorial methods.

Findings

Generated a series expansion for the model.

Derived a linear Fuchsian differential equation of order 4.

Obtained a closed-form expression for the generating function.

Abstract

We have derived the perimeter generating function of a model of punctured staircase polygons in which the internal staircase polygon is rotated by a 90degree angle with respect to the outer staircase polygon. In one approach we calculated a long series expansion for the problem and found that all the terms in the generating function can be reproduced from a linear Fuchsian differential equation of order 4. We then solved this ODE and found a closed form expression for the generating function. This is a highly unusual and most fortuitous result since ODEs of such high order very rarely permit a closed form solution. In a second approach we proved the result for the generating function exactly using combinatorial arguments. This latter solution allows many generalisations including to models with other types of punctures and to a model with any fixed number of nested rotated staircase punctures.