ARITHMETIC PROPERTIES OF COLORED p-ARY PARTITIONS

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).

Automated summary

In this paper, we study the divisibility properties of p-ary partitions colored with k(p - 1) colors, where k is a positive integer. We provide a precise description of the p-adic valuations when k = p(alpha) and k = p(alpha) - 1. Additionally, we prove a general result for the case where finitely many parts can be colored with fewer colors than k(p - 1), while all others require exactly k(p - 1) colors, with k being arbitrary but fixed.