  We are interested in the enumeration of Fully Packed Loop configurations on a
grid with a given noncrossing matching. By the recently proved
Razumov--Stroganov conjecture, these quantities also appear as groundstate
components in the Completely Packed Loop model. When considering matchings with
p nested arches, these numbers are known to be polynomials in p. In this
article, we present several conjectures about these polynomials: in particular,
we describe all real roots, certain values of these polynomials, and conjecture
that the coefficients are positive. The conjectures, which are of a
combinatorial nature, are supported by strong numerical evidence and the proofs
of several special cases. We also give a version of the conjectures when an
extra parameter tau is added to the equations defining the groundstate of the
Completely Packed Loop model.
