On a conjecture of Wilf

Details On a conjecture of Wilf On a conjecture of Wilf

2006-08-03

arxiv.org/abs/math/0608085v2

Journal of Combinatorial Theory, Series A 114 (2007), 1332-1349

Mathematics

Number Theory

18 pages, final version, accepted for publication in the Journal of Combinatorial Theory, Series A

Scientific paper

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 \sum_{j=0}^{n} (-1)^{j} S(n,j) is nonzero for all n>2. We prove this conjecture for all n not congruent to 2 and not congruent to 2944838 modulo 3145728 and discuss applications of this result to graph theory, multiplicative partition functions, and the irrationality of p-adic series.

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