On some polynomials enumerating Fully Packed Loop configurations

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 τ is added to the equations defining the groundstate of the Completely Packed Loop model.