A tiling proof of Euler's pentagnal number theorem and generalizations

TL;DR

This paper extends a combinatorial tiling method to prove Euler's Pentagonal Number Theorem and its generalizations, providing new insights into partition identities through weighted tilings and generating functions.

Contribution

It introduces a key parameter in tiling-based generating functions enabling the proof of Euler's theorem and a broad class of generalizations, expanding the combinatorial approach.

Findings

Recovered Euler's Pentagonal Number Theorem using tiling methods

Established a family of generalizations of the theorem

Enhanced the combinatorial tiling approach with a new parameter

Abstract

In two papers, Little and Sellers introduced an exciting new combinatorial method for proving partition identities which is not directly bijective. Instead, they consider various sets of weighted tilings of a $1 \times \infty$ board with squares and dominoes, and for each type of tiling they construct a generating function in two different ways, which generates a $q$-series identity. Using this method, they recover quite a few classical $q$-series identities, but Euler's Pentagonal Number Theorem is not among them. In this paper, we introduce a key parameter when constructing the generating functions of various sets of tilings which allows us to recover Euler's Pentagonal Number Theorem along with an infinite family of generalizations.