Advertisement

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

  • Dennis Eichhorn
  • Hayan NamEmail author
  • Jaebum Sohn
Article
  • 31 Downloads

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 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.

Keywords

Pentagonal Number Theorem Rank Generalized rank Tiling 

Mathematics Subject Classification

05A17 05A19 11P81 11P84 

Notes

References

  1. 1.
    Alladi, K.: Partition identities involving gaps and weights. Trans. Am. Math. Soc. 349(12), 5001–5019 (1997)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Andrews, G.E.: Generalizations of the Durfee square. J. Lond. Math. Soc (2) 3, 563–570 (1971)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Carlitz, L.: Fibonacci notes. IV. q-Fibonacci polynomials. Fibonacci Quart. 13, 97–102 (1975)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Euler, L.: Introductio in Analysin Infinitorum. Marcum-Michaelum Bousquet, Lausannae (1748)zbMATHGoogle Scholar
  5. 5.
    Little, D.P., Sellers, J.A.: New proofs of identities of Lebesgue and Göllnitz via tilings. J. Comb. Theory Ser. A 116(1), 223–231 (2009)CrossRefGoogle Scholar
  6. 6.
    Little, D.P., Sellers, J.A.: A tiling approach to eight identities of Rogers. Eur. J. Comb. 31(3), 694–709 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Sylvester, J.J.: A constructive theory of partitions, arranged in three acts, an interact and an exodion. Am. J. Math 5, 251–330 (1882)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of California, IrvineIrvineUSA
  2. 2.Yonsei UniversitySeoulSouth Korea

Personalised recommendations