The 2-adic valuations of differences of Stirling numbers of the second kind

Let m, n, k and c be positive integers, ν2(k) be the 2-adic valuation of k and S(n,k) be the Stirling numbers of the second kind. We show that if 2≤m≤n and c is odd, then ν2(S(c2n+1,2m−1)−S(c2n,2m−1))=n+1 except when n=m=2 and c=1, in which case ν2(S(8,3)−S(4,3))=6. This solves a conjecture of Lengyel proposed in 2009.