On The Divisibility By 2 Of The Stirling Numbers Of The Second Kind

In this paper we characterize the divisibility by 2 of the Stirling numbers of the second kind, S(n, k), where n is a sufficiently high power of 2. Let v2(r) denote the highest power of 2 that divides r. We show that there exists a function L(k) such that, for all n > L(k), v2 (k! S(2, k)) = k-l hold, independently from n. (The independence follows from the periodicity of the Stirling numbers modulo any prime power.) For k > 5, the function L(k) can be chosen so that L(k) < k-2. We determine v2{k\S{2 +u,k)) for k>u>\, in particular for u = 1, 2, 3, and 4. We show how to calculate it for negative values, in particular for u 1 . The characterization is generalized for v2(k!S(c-2 + u, k)), where c> 0 denotes an arbitrary odd integer.