Some results on p-adic valuations of Stirling numbers of the second kind

Let n and k be nonnegative integers. The Stirling number of the second kind, denoted by S (n, k), is defined as the number of ways to partition a set of n elements into exactly k nonempty subsets and we have S (n,k) = 1/k! Sigma(k)(i=0)(-1)(i)((k)(i))(k - i)(n). Let p be a prime and nu(p)(n) stand for the p-adic valuation of n, i.e., nu(p()n) is the biggest nonnegative integer r with p(r )dividing n. Divisibility properties of Stirling numbers of the second kind have been studied from a number of different perspectives. In this paper, we present a formula to calculate the exact value of p-adic valuation of S (n, n - k), where n >= k + 1 and 1 <= k <= 7. From this, for any odd prime p, we prove that nu(p) ((n - k)!S (n, n - k)) < n if n >= k + 1 and 0 <= k <= 7. It confirms partially Clarke's conjecture proposed in 1995. We also give some results on nu(p)(S(ap(n), ap(n )- k)), where a and n are positive integers with (a, p) = 1 and 1 <= k <= 7.