Arithmetic properties of colored p-ary partitions

Abstract

We study divisibility properties of p-ary partitions colored with k(p − 1) colors for some positive integer k. In particular, we obtain a precise description of p-adic valuations in the case of \(k=p^{\alpha}\) and \(k=p^{\alpha}-1\).

We also prove a general result concerning the case in which finitely many parts can be colored with a number of colors smaller than k(p − 1) and all others with exactly k(p − 1) colors, where k is arbitrary (but fixed).