Generating function and combinatorial proofs of Elder’s theorem

Robson da Silva, Jorge F. A. Lima, José Plínio O. Santos and Eduardo C. Stabel
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 21, 2015, Number 4, Pages 30—35
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Authors and affiliations

Robson da Silva
ICT, UNIFESP
12247-014, Sao Jose dos Campos-SP, Brazil

Jorge F. A. Lima
IMECC, UNICAMP
C.P. 6065, 13084-970, Campinas-SP, Brazil

José Plínio O. Santos
IMECC, UNICAMP
C.P. 6065, 13084-970, Campinas-SP, Brazil

Eduardo C. Stabel
UFSM
97105-900, Santa Maria-RS, Brazil

Abstract

We revisit Elder’s theorem on integer partitions, which is a generalization of Stanley’s theorem. Two new proofs are presented. The first proof is based on certain tilings of 1 × ∞ boards while the second one is a consequence of a more general identity we prove using generating functions.

Keywords

  • Elder’s theorem
  • Integer partition
  • Generating function
  • Tiling

AMS Classification

  • 11P84
  • 05A19

References

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  8. Schmidt, F. W., & R. Simion (1984) On a Partition Identity, Journal of Combinatorial Theory Series A, Vol. 36, 249–252.
  9. Stabel, E. C. (2011) A Combinatorial Proof of an Identity of Ramanujan Using Tilings. Bulletin of the Brazilian Mathematical Society, 42(2), 203–212.
  10. Stanley, R. P. (1997) Enumerative Combinatorics, Vol. 1, Cambridge University Press, UK.

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Cite this paper

APA

Da Silva, R., Lima, J. F. A., Santos, J. P. O. & Stabel, E. C. (2015). Generating function and combinatorial proofs of Elder’s theorem. Notes on Number Theory and Discrete Mathematics, 21(4), 30-35.

Chicago

Da Silva, Robson, Jorge F. A. Lima, José Plínio O. Santos and Eduardo C. Stabel. “Generating Function and Combinatorial Proofs of Elder’s Theorem.” Notes on Number Theory and Discrete Mathematics 21, no. 4 (2015): 30-35.

MLA

Da Silva, Robson, Jorge F. A. Lima, José Plínio O. Santos and Eduardo C. Stabel. “Generating Function and Combinatorial Proofs of Elder’s Theorem.” Notes on Number Theory and Discrete Mathematics 21.4 (2015): 30-35. Print.

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