The number of fixed points of Wilf's partition involution

Abstract

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, Fill, Janson and Ward (and independently Kane and Rhoades) 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, Janson and Ward also applies to F(n). Specifically, we obtain the asymptotic formula $\log F(n) \sim \frac12 \log f(n)$.