On some polynomials enumerating Fully Packed Loop configurations

TL;DR

This paper investigates polynomials that count Fully Packed Loop configurations with noncrossing matchings, proposing conjectures about their roots, values, and positivity, supported by numerical evidence and special case proofs.

Contribution

It introduces new conjectures about the properties of these polynomials, including their roots and positivity, extending understanding of Fully Packed Loop enumeration.

Findings

Identified all real roots of the polynomials

Conjectured positivity of polynomial coefficients

Supported conjectures with numerical evidence and partial proofs

Abstract

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.