Partitions into parts simultaneously regular, distinct, and/or flat

TL;DR

This paper investigates special classes of integer partitions that are regular, distinct, or flat, analyzing their properties, generating functions, and proposing conjectures about their behavior.

Contribution

It provides new results on the behavior and generating functions of partitions with combined regularity, distinctness, and flatness constraints, including several conjectures.

Findings

Results on partitions that are regular and distinct

Generating functions for partitions with multiple constraints

Conjectures on the behavior of these specialized partitions

Abstract

We explore partitions that lie in the intersection of several sets of classical interest: partitions with parts indivisible by $m$, appearing fewer than $m$ times, or differing by less than $m$. We find results on their behavior and generating functions: more results for those simultaneously regular and distinct, fewest for those distinct and flat. We offer some conjectures in the area.