On some polynomials enumerating Fully Packed Loop configurations

Details On some polynomials enumerating Fully Packed Loop configurations On some polynomials enumerating Fully Packed Loop configurations

2010-02-22

arxiv.org/abs/1002.4187v2

Mathematics

Combinatorics

27 pages. Modifications reflecting the recent proof of the Razumov--Stroganov conjecture; also added some references and a mor

Scientific paper

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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