On the 2-adic order of Stirling numbers of the second kind and their differences

Let n and k be positive integers, d(k) and ν2(k) denote the number of ones in the binary representation of k and the highest power of two dividing k, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that ν2(S(2, k)) = d(k)−1, 1 ≤ k ≤ 2. Here we prove that ν2(S(c2, k)) = d(k)−1, 1 ≤ k ≤ 2, for any positive integer c. We improve and extend this statement in some special cases. For the difference, we obtain lower bounds on ν2 ` S(c2 + u, k)− S(c2 + u, k) ́ for any nonnegative integer u, make a conjecture on the exact order and, for u = 0, prove part of it when k ≤ 6, or k ≥ 5 and d(k) ≤ 2.