Fully Packed Loop configurations in a triangle

TL;DR

This paper explores Fully Packed Loop configurations in a triangular setting, establishing new properties, enumerations, and linear relations that extend understanding beyond existing Razumov--Stroganov correspondence.

Contribution

It introduces and analyzes TFPL configurations, proving their properties, enumerating them under specific conditions, and proposing new linear relations between FPL counts for different grid sizes.

Findings

Proved properties of TFPL configurations.

Enumerated TFPLs under special boundary conditions.

Proposed new linear relations between FPL counts.

Abstract

Fully Packed Loop configurations (FPLs) are certain configurations on the square grid, naturally refined according to certain link patterns. If $A_X$ is the number of FPLs with link pattern $X$, the Razumov--Stroganov correspondence provides relations between numbers $A_X$ relative to a given grid size. In another line of research, if $X\cup p$ denotes $X$ with $p$ additional nested arches, then $A_{X\cup p}$ was shown to be polynomial in $p$: the proof gives rise to certain configurations of FPLs in a triangle (TFPLs). In this work we investigate these TFPL configurations and their relation to FPLs. We prove certain properties of TFPLs, and enumerate them under special boundary conditions. From this study we deduce a class of linear relations, conjectured by Thapper, between quantities $A_X$ relative to different grid sizes, relations which thus differ from the Razumov--Stroganov ones.