The number of fixed points of Wilf's partition involution

TL;DR

This paper derives an asymptotic formula for the number of fixed points of Wilf's partition involution, showing it is approximately the square root of the total number of Wilf partitions.

Contribution

It extends existing asymptotic analysis to the fixed points of Wilf's involution, providing a new formula for their count.

Findings

Asymptotic formula for log F(n) is approximately half of log f(n).

Method from previous work applies to fixed points of the involution.

Provides insight into the structure of Wilf partitions and their involution fixed points.

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 $\log F(n) \sim \frac12 \log f(n)$.