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A Combinatorial Proof of the Sum of qCubes Kristina C. Garrett Department of Mathematics and Computer Science Carleton College, Minnesota, USA kgarrett@carleton.edu Kristen Hummel Department of Mathematics and Computer Science Carleton College, Minnesota, USA hummelk@carleton.edu Submitted: Nov. 2, 2003; Accepted: Dec 15, 2003; Published: Jan. 15, 2004 MR Subject Classifications: 05A17, 05A19 Abstract We give a combinatorial proof of a qanalogue of the classical formula for the sum of cubes. 1 Introduction The classic formula for the sum of the first n cubes, Xn k=1 k3 = n + 1 2 2 , (1) is easily proved by mathematical induction. Many other proofs exist that connect this simple identity to various branches of mathematics. (See [4].) The nature of the right hand side of the identity seems to suggest that a simple combinatorial proof should exist. Indeed, Benjamin and Orrison give such a proof in [2] and other combinatorial proofs are given in [3]. In this paper we will give a qanalogue of (1) and a bijective proof using integer partitions. We begin by reviewing a few of the basics of partition theory. Definition 1.1. An integer partition, , of a positive integer n is a sequence of nonincreasing positive integers = ( 1, 2, . . . , k) such that 1 + 2 + · · · + k = n. The i are the parts of the partition. The number n partitioned by is called the size of the partition and is denoted  . Another method for representing a partition is the graphical representation commonly referred to as the Ferrers shape which was introduced by Sylvester who was writing the electronic journal of combinatorics 10 (2003), #R00 1
Object Description
Collection Title  Scholarly Publications by Carleton Faculty and Staff 
Journal Title  Electronic Journal of Combinatorics 
Article Title  A Combintorial Proof of the Sum of qCubes 
Article Author 
Garrett, Kristina Hummel, Kristin 
Carleton Author 
Garrett, Kristina Hummel, Kristin 
Department  Mathematics 
Field  Science and Mathematics 
Year  2004 
Volume  11 
Publisher  American Mathematical Society 
File Name  009_GarettKristina_ACombinatorialProofOfTheSumOfQCubes.pdf; 009_GarettKristina_ACombinatorialProofOfTheSumOfQCubes.pdf 
Rights Management  This document is authorized for selfarchiving and distribution online by the author(s) and is free for use by researchers. 
RoMEO Color  RoMEO_Color_Green 
Preprint Archiving  Yes 
Postprint Archiving  Yes 
Publisher PDF Archiving  Yes 
Paid OA Option  No_Value 
Fully Open Access  Yes 
Contributing Organization  Carleton College 
Type  Text 
Format  application/pdf 
Language  English 
Description
Article Title  Page 1 
FullText  A Combinatorial Proof of the Sum of qCubes Kristina C. Garrett Department of Mathematics and Computer Science Carleton College, Minnesota, USA kgarrett@carleton.edu Kristen Hummel Department of Mathematics and Computer Science Carleton College, Minnesota, USA hummelk@carleton.edu Submitted: Nov. 2, 2003; Accepted: Dec 15, 2003; Published: Jan. 15, 2004 MR Subject Classifications: 05A17, 05A19 Abstract We give a combinatorial proof of a qanalogue of the classical formula for the sum of cubes. 1 Introduction The classic formula for the sum of the first n cubes, Xn k=1 k3 = n + 1 2 2 , (1) is easily proved by mathematical induction. Many other proofs exist that connect this simple identity to various branches of mathematics. (See [4].) The nature of the right hand side of the identity seems to suggest that a simple combinatorial proof should exist. Indeed, Benjamin and Orrison give such a proof in [2] and other combinatorial proofs are given in [3]. In this paper we will give a qanalogue of (1) and a bijective proof using integer partitions. We begin by reviewing a few of the basics of partition theory. Definition 1.1. An integer partition, , of a positive integer n is a sequence of nonincreasing positive integers = ( 1, 2, . . . , k) such that 1 + 2 + · · · + k = n. The i are the parts of the partition. The number n partitioned by is called the size of the partition and is denoted  . Another method for representing a partition is the graphical representation commonly referred to as the Ferrers shape which was introduced by Sylvester who was writing the electronic journal of combinatorics 10 (2003), #R00 1 