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

**Download 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

- Benjamin, A. T., & J. J. Quinn (2003) Proofs That Really Count: The Art of Combinatorial Proof. The Dolciani Mathematical Expositions, 27.
- Benjamin, A. T., Plott, S., & J. A. Sellers (2008) Tiling Proofs of Recent Sum Identities Involving Pell Numbers. Annals of Combinatorics, 12, 271–278.
- 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.
- 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.
- Hoare, A. H. (1986) An Involution of Blocks in the Partitions of n, The American Mathematical Monthly, 93, 475–476.
- 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.
- Little, D. P., & J. A. Sellers (2010) A Tiling Approach to Eight Identities of Rogers. European Journal of Combinatorics, 31, 694–709.
- Schmidt, F. W., & R. Simion (1984) On a Partition Identity, Journal of Combinatorial Theory Series A, Vol. 36, 249–252.
- Stabel, E. C. (2011) A Combinatorial Proof of an Identity of Ramanujan Using Tilings. Bulletin of the Brazilian Mathematical Society, 42(2), 203–212.
- Stanley, R. P. (1997) Enumerative Combinatorics, Vol. 1, Cambridge University Press, UK.

Related papers

## Cite this paper

APADa 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.

ChicagoDa 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.

MLADa 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.