Enumeration of border-strip decompositions & Weil-Petersson volumes

TL;DR

This paper introduces a permutation-based injection to enumerate border-strip decompositions, providing new formulas and linking these decompositions to Weil-Petersson volumes of moduli spaces.

Contribution

It presents a novel injection from border-strip decompositions to permutations, enabling enumeration and establishing a connection to Weil-Petersson volumes.

Findings

Enumeration formulas for border-strip decompositions

q-analogues of enumeration results

Link between border-strip counts and Weil-Petersson volumes

Abstract

We describe an injection from border-strip decompositions of certain shapes to permutations. This allows us to provide enumeration results, as well as $q$-analogues of enumeration formulas. Finally, we use this injection to prove a connection between the number of border-strip decompositions of the $n\times 2n$ rectangle and the Weil-Petersson volume of the moduli space of an $n$-punctured Riemann sphere.