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 |
|---|---|
| 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
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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 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 -