The p -adic order of Stirling numbers of the second kind

Let n and k be nonnegative integers. The Stirling number of the second kind S(n,k) is defined as the number of ways to partition a set of n elements into exactly k nonempty subsets. Let p be a prime and m be an integer. Define vp(m) to be the p-adic order of m. For 1≤k≤n, if vp(n) is sufficiently large, we deduce the same sharp lower bound for vp(k!S(n,k)) as Adelberg and Lengyel [Citation3] obtained under different assumptions. Furthermore, if k≢n (mod2) and k is divisible by an odd prime p, we get another p-adic estimate of S(n,k), i.e., vp(k!S(n,k))≥vp(n)+2vp(k). These generalize Gessel and Lengyel’s results from (p−1)|n to any other n. Particularly, when p=3, we give the explicit value of v3(k!S(n,k)) that v3(k!S(n,k))={ ⌊k2⌋,if 2∤n, 6∤k and v3(n)≥⌊k2⌋,⌊k−12⌋,if 2|n, 6∤(k−3) and v3(n)≥⌊k−12⌋, which extends the result of Gessel and Lengyel from 2|n to all n quite distinctly from the approach in Adelberg and Lengyel.