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

Abstract: In this work, we put to light a formula that relies the number of fully packed loop configurations (FPLs) associated to a given coupling pi to the number of half-turn symmetric FPLs (HTFPLs) of even size whose coupling is a punctured version of the coupling pi. When the coupling pi is the coupling with all arches parallel pi0 (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. In the case of HTFPLs odd size, and although there is no similar expression, we study the number of HTFPLs whose coupling is a slit version of pi_0, and put to light new puzzling enumerative coincidence involving countings of tilings of hexagons and various symmetry classes of FPLs.