  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.
