A Combinatorial Approach to Sums of Integer Powers

Abstract

n(n + )6(26 + 1), makes this fact plausible even to the beginner and might also spark the conjecture that the resulting polynomial should have degree p + 1. Proofs that validate this or produce a polynomial expression for k'=1,k P often tend to use mathematical induction or recursion. Valuable though they are, such inductive or recursive techniques can sometimes disguise underlying motivation and perhaps leave a reader with a vague feeling of not having grasped the heart of the matter. In this note we offer a combinatorial interpretation of SP(n) that can serve to motivate why this sum can be expressed as a polynomial of degree p + 1 in the variable n. Our approach allows us to make some general statements about coefficients of this polynomial and also produces a technique for direct calculation of the polynomial. We show that calculating this polynomial can be accomplished by solving a ( p 4) X ( p 4) lower triangular system of linear equations. Our exposition requires only the ability to use binomial coefficients in basic counting arguments.