Some useful theorems for asymptotic formulas and their applications to skew plane partitions and cylindric partitions

TL;DR

This paper develops new theorems for asymptotic formulas and applies them to derive asymptotic counts for skew plane and cylindric partitions, revealing different dependencies on shape parameters.

Contribution

It introduces useful theorems for asymptotic analysis and demonstrates their application to skew plane and cylindric partitions, highlighting shape-dependent behaviors.

Findings

Asymptotic formulas depend only on width for skew plane partitions.

Order of asymptotic formulas varies with shape for cylindric partitions.

New theorems improve understanding of partition asymptotics.

Abstract

Inspired by the works of Dewar, Murty and Kot\v{e}\v{s}ovec, we establish some useful theorems for asymptotic formulas. As an application, we obtain asymptotic formulas for the numbers of skew plane partitions and cylindric partitions. We prove that the order of the asymptotic formula for the skew plane partitions of fixed width depends only on the width of the region, not on the profile (the skew zone) itself, while this is not true for cylindric partitions.