Fully Packed Loop configurations in a triangle

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 ∪ p denotes X with p additional nested arches, then A X ∪ p was shown to be polynomial in p: the proof gives rise to certain configurations of FPLs in a triangle (TFPLs). s 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.