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Another proof of the alternating-sign matrix conjecture. (English) Zbl 0859.05027

W. H. Mills, D. P. Robbins and H. Rumsey [J. Comb. Theory, Ser. A 34, 340-359 (1983; Zbl 0516.05016)] made the following conjecture. Theorem 1 (Zeilberger). There are \[ A(n)= {1!4!7!\dots(3n-2)!\over n!(n+1)!(n+2)!\dots (2n-1)!} \] \(n\times n\) alternating sign matrices. Here, an alternating-sign matrix or ASM is a matrix of \(0\)’s, \(1\)’s, and \(-1\)’s such that the nonzero elements in each row and column alternate between \(1\) and \(-1\) and begin and end with \(1\). Alternating sign matrices are related to a number of other combinatorial objects that, remarkably, are also enumerated or conjectured to be enumerated by ratios of progressions of factorials or staggered factorials.
D. Zeilberger [Electron. J. Comb. 3, No. 2, Paper R13, 84 p. (1996; Zbl 0858.05023)] recently proved Theorem 1 by establishing that ASMs are equinumerous with totally symmetric, self-complementary plane partitions, which were enumerated by G. E. Andrews [J. Comb. Theory, Ser. A 66, No. 1, 28-39 (1994; Zbl 0797.05003)]. In this paper, we present a new proof. The most interesting part of the proof is due to A. G. Izergin and V. E. Korepin, who follow Baxter’s remarkable use of the Yang-Baxter equation.

MSC:

05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
15A15 Determinants, permanents, traces, other special matrix functions
05B45 Combinatorial aspects of tessellation and tiling problems
82B23 Exactly solvable models; Bethe ansatz
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