A few months ago Mathematics Magazine published a paper of mine, “A Combinatorial View of Sums of Powers.” In it I give a combinatorial interpretation for the power sum , together with combinatorial proofs of two formulas for this power sum. (An earlier version of some of the results in this paper actually appeared in a blog post from several years ago.)
Recently, however, I received an email from Michael Maltenfort at Northwestern University, pointing out a couple of errors in the paper. I’m recording those errors here.
The most serious mistake Maltenfort pointed out to me is the paper’s claim that m is a nonnegative integer. The formulas actually require m to be a positive integer. For example, in the paper I state the following.
Theorem 1 (Original). The power sum is the number of functions such that, for all , .
However, when this doesn’t work. The power sum as I stated it entails adding up n copies of 1 and so evaluates to n. On the other hand, the set is the set . This means that all functions from to vacuously satisfy the condition in the theorem. The theorem as stated in the paper is off by 1, then, for the case .
Maltenfort suggests changing the lower index on the sum to start at , so that we have the following.
Theorem 1 (Updated). The power sum is the number of functions such that, for all , .
As long as we are fine with the convention that (and I am a fan of this convention in a combinatorial setting), this allows the formula to hold in the case as well.
Maltenfort also points out that the two formulas for the power sum need to be updated in order for them to hold for the case. My paper has
Theorem 2 (Original): , and
Theorem 3 (Original): .
These should be updated to
Theorem 2 (Updated): , and
Theorem 3 (Updated): .
The other mistake in the paper is in the definition of the induced permutation . The definition for in the paper has the requirement
That should instead be
Thanks to Professor Maltenfort for these corrections!
Spivey, Michael Z. “A Combinatorial View of Sums of Powers,” Mathematics Magazine, 90 (2): 125-131, 2021.