Partitions with Distinct Multiplicities of Parts: On An "Unsolved Problem" Posed By Herbert Wilf

TL;DR

This paper investigates Wilf partitions, where all nonzero part multiplicities are distinct, providing asymptotic estimates for their count as the integer size grows.

Contribution

It establishes lead-order asymptotics for the number of Wilf partitions, addressing an open problem posed by Herbert Wilf.

Findings

Derived asymptotic formula for the number of Wilf partitions

Provided insights into the structure of partitions with distinct multiplicities

Addressed a longstanding open problem in partition theory

Abstract

Wilf's Sixth Unsolved Problem asks for any interesting properties of the set of partitions of integers for which the (nonzero) multiplicities of the parts are all different. We refer to these as \emph{Wilf partitions}. Using $f(n)$ to denote the number of Wilf partitions, we establish lead-order asymptotics for $\ln{f(n)}$.