On the class of dominant and subordinate products

TL;DR

This paper introduces new theorems on partition inequalities, demonstrating that parts of size 1 are not necessary, and explores nonnegative power series coefficients of certain rational functions.

Contribution

It provides broad class partition inequalities and illustrates Andrews' anti-telescoping technique, challenging previous assumptions about parts of size 1 in partitions.

Findings

New theorems on partition inequalities

Parts of size 1 are not necessary for irreducible inequalities

Nonnegativity of coefficients in certain rational functions

Abstract

In this paper we provide proofs of two new theorems that provide a broad class of partition inequalities and that illustrate a na\"ive version of Andrews' anti-telescoping technique quite well. These new theorems also put to rest any notion that including parts of size 1 is somehow necessary in order to have a valid irreducible partition inequality. In addition, we prove (as a lemma to one of the theorems) a rather nontrivial class of rational functions of three variables has entirely nonnegative power series coefficients.