On the link pattern distribution of quarter-turn symmetric FPL configurations

TL;DR

This paper investigates the distribution of link patterns in quarter-turn symmetric fully-packed loop configurations, proposing new conjectures, proving a special case, and deriving a formula related to plane partitions.

Contribution

It introduces new conjectures on link pattern distributions, proves a specific case, and connects these findings to enumeration formulas for quasi-symmetric plane partitions.

Findings

Proposed new conjectures on link pattern distributions.

Proved a special case confirming the conjectured probability.

Derived a formula for counting a class of quasi-symmetric plane partitions.

Abstract

We present new conjectures on the distribution of link patterns for fully-packed loop (FPL) configurations that are invariant, or almost invariant, under a quarter turn rotation, extending previous conjectures of Razumov and Stroganov and of de Gier. We prove a special case, showing that the link pattern that is conjectured to be the rarest does have the prescribed probability. As a byproduct, we get a formula for the enumeration of a new class of quasi-symmetry of plane partitions.