Partitions into three triangular numbers

Michael D. Hirschhorn, James Allen Sellers

Research output: Contribution to journalArticle

2 Citations (Scopus)

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(n), the number of partitions of the integer n into three triangular numbers, as well as pd (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(n) and pd (n) appear to satisfy very few arithmetic relations (apart from certain parity results). However, we shall show that, for all n ≥ 0, p(27n + 12) = 3p(3n + 1) and pd (27n + 12) = 3pd (3n + 1). Two separate proofs of these results are given, one via generating function manipulations and the other by a combinatorial argument.

Original languageEnglish (US)
Pages (from-to)307-318
Number of pages12
JournalAustralasian Journal of Combinatorics
Volume30
StatePublished - 2004

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Triangular number
Partition
Combinatorial argument
Integer
Parity
Gauss
Generating Function
Manipulation
Count
Distinct

All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics

Cite this

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

In: Australasian Journal of Combinatorics, Vol. 30, 2004, p. 307-318.

Research output: Contribution to journalArticle

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