Alternating sign matrices and descending plane partitions

Square ice, alternating sign matrices, and classical orthogonal polynomials

“…[4] is readily recovered. Moreover, recalling (3.8), and specializing λ = π/2, that is φ = π/2 or tan(φ/2) = 1, the expression (3.6) for the refined 2-enumeration, obtained in [7,12,13,15], is immediately reproduced.…”

“…The answer in the case of 3-enumeration, again conjectured in [6,7], was subsequently [10] proved to be…”

“…The extension of the x = 1 result above to the case of generic x is not known, but for a few nontrivial cases, namely x = 2 and x = 3, closed expressions for x-enumerations are known (the case x = 0 is trivial since assigning a vanishing weight to each −1 entry restricts the enumeration to the sole permutation matrices: A(n; 0) = n!). The result in the case x = 2, related to the domino tilings of Aztec diamond [12,13,15], have been obtained in [7], and reads…”

“…The standard notation for the refined x-enumeration is A(N, r; x); in the case x = 1 one writes simply A(N, r) just like A(N) for the total number of ASMs. It was again conjectured in [6,7], and proved in [23] that refined enumeration of ASMs is given by…”

“…The model, in its inhomogeneous formulation, i.e., with position-dependent Boltzmann weights, was originally proposed within the theory of correlation functions of quantum integrable models, in the framework of the quantum inverse scattering method [5]. The model was later found to be deeply related with the problems of enumeration of alternating sign matrices (ASMs) [6][7][8][9][10][11] and domino tilings [12][13][14][15]. It should be mentioned that ASM enumerations appear to be in turn deeply related with quantum spin chains and some loop models, via Razumov-Stroganov conjecture [16]; for recent works, see for instance Ref.…”