Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: A Combinatorial Proof of the Sum of q-Cubes. | A Combinatorial Proof of the Sum of Ợ-Cubes Kristina C. Garrett Department of Mathematics and Computer Science Carleton College Minnesota USA kgarrett@ Kristen Hummel Department of Mathematics and Computer Science Carleton College Minnesota USA hummelk@ Submitted Nov 2 2003 Accepted Dec 15 2003 Published Jan 23 2004 MR Subject Classifications 05A17 05A19 Abstract We give a combinatorial proof of a ợ-analogue of the classical formula for the sum of cubes. 1 Introduction The classic formula for the sum of the first n cubes n X k k 1 I 1 2 n 1 2 J 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 -analogue of 1 and a bijective proof using integer partitions. We begin by reviewing a few of the basics of partition theory. Definition . An integer partition A of a positive integer n is a sequence of nonincreasing positive integers A Al A2 . Ak such that Al A2 Ak n. The Aị are the parts of the partition. The number n partitioned by A is called the size of the partition and is denoted A . Another method for representing a partition A is the graphical representation commonly referred to as the Ferrers shape which was introduced by Sylvester who was writing THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 R9 1 about a proof described to him by . Ferrers. The Ferrers shape of a partition is an array of boxes left justified in which the number of boxes in the first row is equal to the size of the first part the number of boxes in the second row is equal to the size of the second part etc. For example the Ferrers diagram for the partition A 4 4 3 2 1 is shown in Figure 1. Figure 1 Ferrers Diagram for A 4 4 3 2 1 . .
