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
Full paper (PDF, 208 Kb)

Details

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

1. Benjamin, A. T., & J. J. Quinn (2003) Proofs That Really Count: The Art of Combinatorial Proof. The Dolciani Mathematical Expositions, 27.
2. Benjamin, A. T., Plott, S., & J. A. Sellers (2008) Tiling Proofs of Recent Sum Identities Involving Pell Numbers. Annals of Combinatorics, 12, 271–278.
3. Briggs, K. S., Little, D. P., & J. A. Sellers (2011) Combinatorial Proofs of Various q-Pell Identities via Tilings. Annals of Combinatorics, 14(4), 407–418.
4. Dastidar, M. G., & S. S. Gupta (2013) Generalization of a few results in integer partitions. Notes on Number Theory and Discrete Mathematics, 19(2), 69–76.
5. Hoare, A. H. (1986) An Involution of Blocks in the Partitions of n, The American Mathematical Monthly, 93, 475–476.
6. Kirdar, M. S., & T. H. R. Skyrme (1982) On an Identity Related to Partitions and Repetitions of Parts, Canadian Journal of Mathematics, 34, 194–195.
7. Little, D. P., & J. A. Sellers (2010) A Tiling Approach to Eight Identities of Rogers. European Journal of Combinatorics, 31, 694–709.
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.

Related papers