On a conjecture of Wilf

Authors:
Stefan De Wannemacker;Thomas Laffey;Robert Osburn
Affiliations:
Faculteit Wetenschappen, K.U. Leuven Campus Kortrijk, E. Sabbelaan 53, B-8500 Kortrijk, Belgium;School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland;School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland
Venue:
Journal of Combinatorial Theory Series A
Year:
2007

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Abstract

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=0n(-1)^jS(n,j) is nonzero for all n2. We prove this conjecture for all n@?2 and @?2944838mod3145728 and discuss applications of this result to graph theory, multiplicative partition functions, and the irrationality of p-adic series.