Half-turn symmetric FPLs with rare couplings and tilings of hexagons

Abstract

We study a formula that links the number of fully packed loop configurations (FPLs) associated to a given coupling π to the number of half-turn symmetric FPLs (HTFPLs) of even size whose coupling is a punctured version of the coupling π. When the coupling π is the coupling with all arches parallel π0 (the “rarest” one), this formula states the equality of the number of corresponding HTFPLs to the number of cyclically-symmetric plane partition of the same size. We provide a bijective proof of this fact. While there is no similar expression for HTFPLs of odd size, we study the number of HTFPLs whose coupling is a slit version of π0, and discover new puzzling enumerative coincidences involving countings of tilings of hexagons and various symmetry classes of FPLs.