On the weighted enumeration of alternating sign matrices and descending plane partitions

Abstract

We prove a conjecture of Mills, Robbins and Rumsey [Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (1983), 340-359] that, for any n, k, m and p, the number of n × n alternating sign matrices (ASMs) for which the 1 of the first row is in column k + 1 and there are exactly m −1's and m + p inversions is equal to the number of descending plane partitions (DPPs) for which each part is at most n and there are exactly k parts equal to n, m special parts and p nonspecial parts. The proof involves expressing the associated generating functions for ASMs and DPPs with fixed n as determinants of n × n matrices, and using elementary transformations to show that these determinants are equal. The determinants themselves are obtained by standard methods: for ASMs this involves using the Izergin-Korepin formula for the partition function of the six-vertex model with domain-wall boundary conditions, together with a bijection between ASMs and configurations of this model, and for DPPs it involves using the Lindström-Gessel-Viennot theorem, together with a bijection between DPPs and certain sets of nonintersecting lattice paths.

Integrable Combinatorics

“…(This is obtained by letting all z i → z and all w i → w, with a = a(z, w), b = b(z, w) and c = c(z, w).) This and more refinements were worked out in [3]. We have the following remarkable result:…”

“…The complete MRR conjecture is now proved [3]. The proof uses the same ingredients as exposed here, and includes extra refinements.…”

Arctic Curves of the Twenty-Vertex Model with Domain Wall Boundaries

“…To derive such a curve, it would be desirable to have more explicit expressions for the model partition function in terms of spectral parameters, giving access to boundary one-point functions. 12 The fact that the arctic curve of the Holey Hexagon model and that of the uniform 6V model share a common portion is a direct consequence of the refined ASM-DPP correspondence shown in [BDFZJ12].…”

“…Remarkably, we find exactly the same pattern of correspondences if we compare the arctic curve for cyclically symmetric rhombus tilings of a Holey Hexagon, in bijection with descending plane partitions (DPP) [Kra06] to that of the uniform 6V model (with weights a = b = c) with DWBC, in bijection with Alternating Sign Matrices (ASM). The ASM-DPP correspondence was proved with its highest level of refinement in [BDFZJ12,BDFZJ13]. In the Holey Hexagon model, the tiled domain is now formed of a fundamental domain with a rhombic shape drawn on the triangular lattice, and two extra copies of this domain obtained by two 120 • rotations For the Holey Hexagon rhombus tiling, the remaining part is the analytic continuation of this shared portion, forming an ellipse.…”

The Ground-State Vector of the XY Heisenberg Chain and the Gauss Decomposition

“…The correlation functions of certain quantum integrable models demonstrate connection with enumerative combinatorics and with the theory of symmetric functions [1][2][3][4][5]. For instance, random lattice walks and boxed plane partitions, as subjects of enumerative combinatorics [6], are related to the correlation functions of the XX model [7][8][9][10][11].…”