On the p-adic valuations of some special Stirling numbers of the second kind

Abstract:

Let k and n be positive integers. The Stirling numbers of the second kind is defined as the number of ways to partition a set of n elements into exactly k non-empty subsets, denoted by S(n,k). Given a prime p and a positive integer n, there exist unique integers a and m, with m≥0 and (a,p)=1, such that n=ap^m. The number m is called p-adic valuation of n, denoted by v_p(n)=m. The study of p-adic valuations of Stirling numbers of the second kind is important in number theory and algebraic topology, and full with challenging problems. In this paper, we study the p-adic valuations of the Stirling numbers of the second kind with the special form S(p^n,p2^t) when t and n are positive integers. Let p be an odd prime. We show that if n≥2 and t≥1, then v_p(S(p^n,p2^t))≥n+2-2^t. This extends the results obtained by Zhao and Qiu recently.