2010/02/22 by Tiago Fonseca, Fonseca, Tiago, Philippe Nadeau +1
Mathematics · Physics and Astronomy · #05A05 #05A17 #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.CO #math.MP #msc:05A05 #msc:05A17
paper · pdf · doi:10.48550/arxiv.1002.4187
27 pages. Modifications reflecting the recent proof of the Razumov--Stroganov conjecture; also added some references and a more detailed conclusion
arxiv created 2010/06/21 · arxiv updated 2010/06/22
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.