### 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 language | English |
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Journal | Ramanujan Journal |

DOIs | |

Publication status | Accepted/In press - 2019 Jan 1 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Ramanujan Journal*. https://doi.org/10.1007/s11139-019-00189-2

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - A tiling proof of Euler’s Pentagonal Number Theorem and generalizations

AU - Eichhorn, Dennis

AU - Nam, Hayan

AU - Sohn, Jaebum

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85075249765&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85075249765&partnerID=8YFLogxK

U2 - 10.1007/s11139-019-00189-2

DO - 10.1007/s11139-019-00189-2

M3 - Article

AN - SCOPUS:85075249765

JO - Ramanujan Journal

JF - Ramanujan Journal

SN - 1382-4090

ER -