The number of fixed points of Wilf's partition involution (Q396924)

Summary: Wilf partitions are partitions of an integer \(n\) in which all nonzero multiplicities are distinct. On his webpage, the late Herbert Wilf posed the problem to find ``any interesting theorems'' about the number \(f(n)\) of those partitions. Recently, \textit{J. A. Fill} et al. [Electron. J. Comb. 19, No. 2, Research Paper P18, 6 p. (2012; Zbl 1243.05025)] (and independently \textit{D. Kane} and \textit{R. C. Rhoades} [``Asymptotics for Wilf's partitions with distinct multiplicities'', Preprint (2013)]) determined an asymptotic formula for \(\log f(n)\). Since the original motivation for studying Wilf partitions was the fact that the operation that interchanges part sizes and multiplicities is an involution on the set of Wilf partitions, they mentioned as an open problem to determine a similar asymptotic formula for the number of fixed points of this involution, which we denote by \(F(n)\). In this short note, we show that the method of Fill et al. [loc. cit.] also applies to \(F(n)\). Specifically, we obtain the asymptotic formula \(\log F(n) \sim \frac12 \log f(n)\).