This paper examines conditions under which \(\sum_{n=1}^k n^tx^k\) is a permutation polynomial over a field of prime order \(p\). By applying Hermite's criterion, it then deduces a very particular congruence mod \(p\) which involves compositions (also known as ordered partitions) of multiples of \(p-1\). This result generalises an earlier result from a paper by the second author and \textit{M. K. Singh} [Integers 18, Paper A73, 6 p. (2018; Zbl 1459.11207)].