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 tau is added to the equations defining the groundstate of the Completely Packed Loop model.Comment: 27 pages. Modifications reflecting the recent proof of the Razumov--Stroganov conjecture; also added some references and a more detailed conclusio

Fully Packed O(n=1) Model on Random Eulerian Triangulations

We introduce a matrix model describing the fully-packed O(n) model on random Eulerian triangulations (i.e. triangulations with all vertices of even valency). For n=1 the model is mapped onto a particular gravitational 6-vertex model with central charge c=1, hence displaying the expected shift c -> c+1 when going from ordinary random triangulations to Eulerian ones. The case of arbitrary n is also discussed.Comment: 12 pages, 3 figures, tex, harvmac, eps

Higher Spin Alternating Sign Matrices

We define a higher spin alternating sign matrix to be an integer-entry square matrix in which, for a nonnegative integer r, all complete row and column sums are r, and all partial row and column sums extending from each end of the row or column are nonnegative. Such matrices correspond to configurations of spin r/2 statistical mechanical vertex models with domain-wall boundary conditions. The case r=1 gives standard alternating sign matrices, while the case in which all matrix entries are nonnegative gives semimagic squares. We show that the higher spin alternating sign matrices of size n are the integer points of the r-th dilate of an integral convex polytope of dimension (n-1)^2 whose vertices are the standard alternating sign matrices of size n. It then follows that, for fixed n, these matrices are enumerated by an Ehrhart polynomial in r.Comment: 41 pages; v2: minor change

On the link pattern distribution of quarter-turn symmetric FPL configurations

We present new conjectures on the distribution of link patterns for fully-packed loop (FPL) configurations that are invariant, or almost invariant, under a quarter turn rotation, extending previous conjectures of Razumov and Stroganov and of de Gier. We prove a special case, showing that the link pattern that is conjectured to be the rarest does have the prescribed probability. As a byproduct, we get a formula for the enumeration of a new class of quasi-symmetry of plane partitions.Comment: 12 pages, 6 figures. Submitted to FPSAC 200

Fully Packed Loops in a triangle: matchings, paths and puzzles

Fully Packed Loop configurations in a triangle (TFPLs) first appeared in the study of ordinary Fully Packed Loop configurations (FPLs) on the square grid where they were used to show that the number of FPLs with a given link pattern that has m nested arches is a polynomial function in m. It soon turned out that TFPLs possess a number of other nice properties. For instance, they can be seen as a generalized model of Littlewood-Richardson coefficients. We start our article by introducing oriented versions of TFPLs; their main advantage in comparison with ordinary TFPLs is that they involve only local constraints. Three main contributions are provided. Firstly, we show that the number of ordinary TFPLs can be extracted from a weighted enumeration of oriented TFPLs and thus it suffices to consider the latter. Secondly, we decompose oriented TFPLs into two matchings and use a classical bijection to obtain two families of nonintersecting lattice paths (path tangles). This point of view turns out to be extremely useful for giving easy proofs of previously known conditions on the boundary of TFPLs necessary for them to exist. One example is the inequality d(u)+d(v)<=d(w) where u,v,w are 01-words that encode the boundary conditions of ordinary TFPLs and d(u) is the number of cells in the Ferrers diagram associated with u. In the third part we consider TFPLs with d(w)- d(u)-d(v)=0,1; in the first case their numbers are given by Littlewood-Richardson coefficients, but also in the second case we provide formulas that are in terms of Littlewood-Richardson coefficients. The proofs of these formulas are of a purely combinatorial nature.Comment: 40 pages, 31 figure

Determinant Formulae for some Tiling Problems and Application to Fully Packed Loops

We present determinant formulae for the number of tilings of various domains in relation with Alternating Sign Matrix and Fully Packed Loop enumeration