### Abstract

A celebrated result of Gauss states that every positive integer can be represented as the sum of three triangular numbers. In this article we study p_{3Δ}(n), the number of partitions of the integer n into three triangular numbers, as well as p^{d}
_{3Δ}(n), the number of partitions of n into three distinct triangular numbers. Unlike t(n), which counts the number of representations of n into three triangular numbers, p_{3Δ}(n) and p^{d}
_{3Δ}(n) appear to satisfy very few arithmetic relations (apart from certain parity results). However, we shall show that, for all n ≥ 0, p_{3Δ}(27n + 12) = 3p_{3Δ}(3n + 1) and p^{d}
_{3Δ}(27n + 12) = 3p^{d}
_{3Δ}(3n + 1). Two separate proofs of these results are given, one via generating function manipulations and the other by a combinatorial argument.

Original language | English (US) |
---|---|

Pages (from-to) | 307-318 |

Number of pages | 12 |

Journal | Australasian Journal of Combinatorics |

Volume | 30 |

State | Published - 2004 |

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### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics

### Cite this

*Australasian Journal of Combinatorics*,

*30*, 307-318.

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*Australasian Journal of Combinatorics*, vol. 30, pp. 307-318.

**Partitions into three triangular numbers.** / Hirschhorn, Michael D.; Sellers, James Allen.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Partitions into three triangular numbers

AU - Hirschhorn, Michael D.

AU - Sellers, James Allen

PY - 2004

Y1 - 2004

N2 - A celebrated result of Gauss states that every positive integer can be represented as the sum of three triangular numbers. In this article we study p3Δ(n), the number of partitions of the integer n into three triangular numbers, as well as pd 3Δ(n), the number of partitions of n into three distinct triangular numbers. Unlike t(n), which counts the number of representations of n into three triangular numbers, p3Δ(n) and pd 3Δ(n) appear to satisfy very few arithmetic relations (apart from certain parity results). However, we shall show that, for all n ≥ 0, p3Δ(27n + 12) = 3p3Δ(3n + 1) and pd 3Δ(27n + 12) = 3pd 3Δ(3n + 1). Two separate proofs of these results are given, one via generating function manipulations and the other by a combinatorial argument.

AB - A celebrated result of Gauss states that every positive integer can be represented as the sum of three triangular numbers. In this article we study p3Δ(n), the number of partitions of the integer n into three triangular numbers, as well as pd 3Δ(n), the number of partitions of n into three distinct triangular numbers. Unlike t(n), which counts the number of representations of n into three triangular numbers, p3Δ(n) and pd 3Δ(n) appear to satisfy very few arithmetic relations (apart from certain parity results). However, we shall show that, for all n ≥ 0, p3Δ(27n + 12) = 3p3Δ(3n + 1) and pd 3Δ(27n + 12) = 3pd 3Δ(3n + 1). Two separate proofs of these results are given, one via generating function manipulations and the other by a combinatorial argument.

UR - http://www.scopus.com/inward/record.url?scp=80052881143&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80052881143&partnerID=8YFLogxK

M3 - Article

VL - 30

SP - 307

EP - 318

JO - Australasian Journal of Combinatorics

JF - Australasian Journal of Combinatorics

SN - 1034-4942

ER -