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arXiv · 1002.4187

On some polynomials enumerating Fully Packed Loop configurations

Tiago Fonseca·Philippe Nadeau
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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.

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Tiago Fonseca, Philippe Nadeau. 2010-02-22. On some polynomials enumerating Fully Packed Loop configurations. https://arxiv.org/abs/1002.4187

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