On a square-ice analogue of plane partitions

TL;DR

This paper investigates a new class of square-ice configurations analogous to plane partitions, using enumeration and simulation methods to analyze their asymptotic behavior and establish similarities with classical plane partitions.

Contribution

It introduces a one-parameter family of square-ice analogues of plane partitions and analyzes their asymptotic properties through enumeration and Monte Carlo simulations.

Findings

Asymptotic behavior similar to plane partitions

Successful enumeration and simulation of configurations

Provided independent estimates for asymptotics

Abstract

We study a one-parameter family ($\ell=1,2,3,\ldots$) of configurations that are square-ice analogues of plane partitions. Using an algorithm due to Bratley and McKay, we carry out exact enumerations in order to study their asymptotic behaviour and establish, via Monte Carlo simulations as well as explicit bounds, that the asymptotic behaviour is similar to that of plane partitions. We finally carry out a series analysis and provide independent estimates for the asymptotic behaviour.