A note on multiset reconstruction from sum and pairwise products

TL;DR

This paper proves a conjecture that the map from integer partitions to their pairwise products uniquely determines the original partition, confirming injectivity for all cases.

Contribution

It provides a proof that the pre_2 map is injective on the set of integer partitions of any integer n, resolving a conjecture by Ballantine, Beck, Merca, and Sagan.

Findings

The pre_2 map is injective on integer partitions of any n.

The conjecture by Ballantine, Beck, Merca, and Sagan is confirmed.

The result advances understanding of partition reconstruction from pairwise products.

Abstract

Ballantine, Beck, and Merca defined a map $\mathrm{pre}_2$, which sends an integer partition $\lambda = (\lambda_1, \dots, \lambda_{\ell})$ to the set $\{\lambda_i\lambda_j : 1 \leq i < j \leq \ell\}$ consisting of the pairwise products of parts of $\lambda$. The same three authors and Sagan conjectured that for each $n$, the map $\mathrm{pre}_2$ is injective on the set of integer partitions of $n$. In this note, we prove their conjecture.