On FPL configurations with four sets of nested arches

TL;DR

This paper explores the enumeration of Fully Packed Loop configurations with four nested arch sets, linking it to tiling problems and deriving explicit determinant formulas for specific cases.

Contribution

It introduces a novel approach to count FPL configurations with four nested arch sets using tiling and non-intersecting line techniques, providing explicit formulas.

Findings

Derived explicit formulas for configurations with min(a,b,c,d)=1 or 2.

Established a determinant formula for configurations with b=d.

Connected FPL enumeration to tiling problems on the triangular lattice.

Abstract

The problem of counting the number of Fully Packed Loop (FPL) configurations with four sets of a,b,c,d nested arches is addressed. It is shown that it may be expressed as the problem of enumeration of tilings of a domain of the triangular lattice with a conic singularity. After reexpression in terms of non-intersecting lines, the Lindstr\"om-Gessel-Viennot theorem leads to a formula as a sum of determinants. This is made quite explicit when min(a,b,c,d)=1 or 2. We also find a compact determinant formula which generates the numbers of configurations with b=d.