On fully packed loop configurations with four sets of nested arches

Abstract

The problem of counting the number of fully packed loop (FPL) configurations with four sets ofa, b, c, d nested arches is addressed. It is shown that it may be expressed as the problem ofenumeration of tilings of a domain of the triangular lattice with a conic singularity. Afterre-expression in terms of non-intersecting lines, the Lindström–Gessel–Viennot theoremleads to a formula as a sum of determinants. This is made quite explicit whenmin(a, b, c, d) = 1 or 2. We also find a compact determinant formula which generates the numbers of configurations withb = d.