On fully packed loop configurations with four sets of nested arches

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 re-expression in terms of non-intersecting lines, the Lindström-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.