On the positivity of infinite products connected to partitions with even parts below odd parts and copartitions

TL;DR

This paper provides a combinatorial proof of positivity results related to partitions and copartitions, offering new perspectives and extending to broader positivity conjectures with partial proofs.

Contribution

It introduces a combinatorial approach to prove positivity of certain infinite products connected to partitions and copartitions, and proposes new conjectures.

Findings

Combinatorial proof of Chern's positivity result

Reproof of an overpartition result using copartitions

Formulation of new positivity conjectures for infinite and finite products

Abstract

In this paper, we give a combinatorial proof of a positivity result of Chern related to Andrews's $\mathcal{EO}^*$-type partitions. This combinatorial proof comes after reframing Chern's result in terms of copartitions. Using this new perspective, we also reprove an overpartition result of Chern by showing that it comes essentially "for free" from our combinatorial proof and some basic properties of copartitions. Finally, the application of copartitions leads us to more general positivity conjectures for families of both infinite and finite products, with a proof in one special case.