Enumeration of alternating sign matrices of even size (quasi)-invariant under a quarter-turn rotation

The aim of this work is to enumerate alternating sign matrices (ASM) that are quasi-invariant under a quarter-turn. The enumeration formula (conjectured by Duchon) involves, as a product of three terms, the number of unrestricted ASM's and the number of half-turn symmetric ASM's

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

Multiply-refined enumeration of alternating sign matrices

Four natural boundary statistics and two natural bulk statistics are considered for alternating sign matrices (ASMs). Specifically, these statistics are the positions of the 1's in the first and last rows and columns of an ASM, and the numbers of generalized inversions and -1's in an ASM. Previously-known and related results for the exact enumeration of ASMs with prescribed values of some of these statistics are discussed in detail. A quadratic relation which recursively determines the generating function associated with all six statistics is then obtained. This relation also leads to various new identities satisfied by generating functions associated with fewer than six of the statistics. The derivation of the relation involves combining the Desnanot-Jacobi determinant identity with the Izergin-Korepin formula for the partition function of the six-vertex model with domain-wall boundary conditions.Comment: 62 pages; v3 slightly updated relative to published versio

On the symmetry of the partition function of some square ice models

We consider the partition function Z(N;x_1,...,x_N,y_1,...,y_N) of the square ice model with domain wall boundary. We give a simple proof of the symmetry of Z with respect to all its variables when the global parameter a of the model is set to the special value a=exp(i\pi/3). Our proof does not use any determinantal interpretation of Z and can be adapted to other situations (for examples to some symmetric ice models).Comment: 8 page

Extreme diagonally and antidiagonally symmetric alternating sign matrices of odd order

For each $\alpha \in \{0,1,-1 \}$, we count diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order with a maximal number of $\alpha$'s along the diagonal and the antidiagonal, as well as DASASMs of fixed odd order with a minimal number of $0$'s along the diagonal and the antidiagonal. In these enumerations, we encounter product formulas that have previously appeared in plane partition or alternating sign matrix counting, namely for the number of all alternating sign matrices, the number of cyclically symmetric plane partitions in a given box, and the number of vertically and horizontally symmetric ASMs. We also prove several refinements. For instance, in the case of DASASMs with a maximal number of $-1$'s along the diagonal and the antidiagonal, these considerations lead naturally to the definition of alternating sign triangles. These are new objects that are equinumerous with ASMs, and we are able to prove a two parameter refinement of this fact, involving the number of $-1$'s and the inversion number on the ASM side. To prove our results, we extend techniques to deal with triangular six-vertex configurations that have recently successfully been applied to settle Robbins' conjecture on the number of all DASASMs of odd order. Importantly, we use a general solution of the reflection equation to prove the symmetry of the partition function in the spectral parameters. In all of our cases, we derive determinant or Pfaffian formulas for the partition functions, which we then specialize in order to obtain the product formulas for the various classes of extreme odd DASASMs under consideration.Comment: 41 pages, several minor improvements in response to referee's comments. Final version. Matches published version except for very minor change

Half-turn symmetric FPLs with rare couplings and tilings of hexagons

14 p.International audienceIn this work, we put to light a formula that relies the number of fully packed loop configurations (FPLs) associated to a given coupling pi to the number of half-turn symmetric FPLs (HTFPLs) of even size whose coupling is a punctured version of the coupling pi. When the coupling pi is the coupling with all arches parallel pi0 (the ''rarest'' one), this formula states the equality of the number of corresponding HTFPLs to the number of cyclically-symmetric plane partition of the same size. We provide a bijective proof of this fact. In the case of HTFPLs odd size, and although there is no similar expression, we study the number of HTFPLs whose coupling is a slit version of pi_0, and put to light new puzzling enumerative coincidence involving countings of tilings of hexagons and various symmetry classes of FPLs