On a conjecture of Wilf

Let n and k be natural numbers and let S ( n , k ) denote the Stirling numbers of the second kind. It is a conjecture of Wilf that the alternating sum ∑ j = 0 n ( − 1 ) j S ( n , j ) is nonzero for all n > 2 . We prove this conjecture for all n ≢ 2 and ≢ 2 944 838 mod 3 145 728 and discuss applications of this result to graph theory, multiplicative partition functions, and the irrationality of p-adic series.