Two Questions about the Fractional Counting of Partitions

TL;DR

This paper discusses fractional enumeration of integer partitions, raises two questions about fractional counting, and announces that both questions have been answered by other researchers, with donations made to OEIS.

Contribution

It introduces two open questions about fractional counting of partitions and reports their solutions by other researchers, advancing understanding in fractional enumeration.

Findings

Both questions about fractional counting of partitions have been answered.

Solutions were provided by Will Sawin and Christopher Ryba.

Donations made to OEIS in honor of the researchers.

Abstract

We recall the notion of fractional enumeration and immediately focus on the fractional counting of integer partitions, where each partition gets credit equal to the reciprocal of the product of its parts. We raise two intriguing questions regarding this count, and for each of these questions, we are pledging a 100 US dollars donation to the OEIS in honor of the first solver. In this revised version we announce that both questions have been answered. The first was first answered by Will Sawin and the second was answered by Christopher Ryba. A donation to the OEIS, of 100 US dollars each, in their honor has been made.