A tiling proof of Euler’s Pentagonal Number Theorem and generalizations

Dennis Eichhorn, Hayan Nam, Jaebum Sohn

Research output: Contribution to journalArticle

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 × ∞ board with squares and dominoes, and for each type of tiling they construct a generating function in two different ways, which in turn 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 uncountably infinite family of generalizations.

Original languageEnglish
JournalRamanujan Journal
DOIs
Publication statusAccepted/In press - 2019 Jan 1

Fingerprint

Pentagonal number
Tiling
Q-series
Generating Function
Theorem
Partition Identities
Bijective
Generalization

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Cite this

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A tiling proof of Euler’s Pentagonal Number Theorem and generalizations. / Eichhorn, Dennis; Nam, Hayan; Sohn, Jaebum.

In: Ramanujan Journal, 01.01.2019.

Research output: Contribution to journalArticle

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