On p-adic valuations of certain m colored p-ary partition functions

Abstract

AbstractLet $$k\in \mathbb {N}_{\ge 2}$$ k ∈ N ≥ 2 and for given $$m\in \mathbb {Z}{\setminus }\{0\}$$ m ∈ Z \ { 0 } consider the sequence $$(S_{k,m}(n))_{n\in \mathbb {N}}$$ ( S k , m ( n ) ) n ∈ N defined by the power series expansion $$\begin{aligned} \frac{1}{(1-x)^{m}}\prod _{i=0}^{\infty } \frac{1}{(1-x^{k^{i}})^{m(k-1)}}=\sum _{n=0}^{\infty }S_{k,m}(n)x^{n}. \end{aligned}$$ 1 ( 1 - x ) m ∏ i = 0 ∞ 1 ( 1 - x k i ) m ( k - 1 ) = ∑ n = 0 ∞ S k , m ( n ) x n . The number $$S_{k,m}(n)$$ S k , m ( n ) for $$m\in \mathbb {N}_{+}$$ m ∈ N + has a natural combinatorial interpretation: it counts the number of representations of n as sums of powers of k, where the part equal to 1 takes one among mk colors and each part $$>1$$ > 1 takes $$m(k-1)$$ m ( k - 1 ) colors. We concentrate on the case when $$k=p$$ k = p is a prime. Our main result is the computation of the exact value of the p-adic valuation of $$S_{p,m}(n)$$ S p , m ( n ) . In particular, in each case the set of values of $$\nu _{p}(S_{p,m}(n))$$ ν p ( S p , m ( n ) ) is finite and the maximum value is bounded by $${\text {max}}\{\nu _{p}(m)+1,\nu _{p}(m+1)+1\}$$ max { ν p ( m ) + 1 , ν p ( m + 1 ) + 1 } . Our results can be seen as a generalization of earlier work of Churchhouse and recent work of Gawron, Miska and Ulas, and the present authors.