On a conjecture of Wilf

Let n and k   be natural numbers and let S(n,k)S(n,k) denote the Stirling numbers of the second kind. It is a conjecture of Wilf that the alternating sum∑j=0n(−1)jS(n,j) is nonzero for all n>2n>2. We prove this conjecture for all n≢2n≢2 and ≢2944838mod3145728 and discuss applications of this result to graph theory, multiplicative partition functions, and the irrationality of p-adic series.