Kuperberg, Greg Another proof of the alternating-sign matrix conjecture. (English) Zbl 0859.05027 Int. Math. Res. Not. 1996, No. 3, 139-150 (1996). 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. Cited in 1 ReviewCited in 177 Documents 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 Keywords:alternating-sign matrix Citations:Zbl 0516.05016; Zbl 0797.05003; Zbl 0858.05023 PDFBibTeX XMLCite \textit{G. Kuperberg}, Int. Math. Res. Not. 1996, No. 3, 139--150 (1996; Zbl 0859.05027) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: Robbins numbers: a(n) = Product_{k=0..n-1} (3k+1)!/(n+k)!; also the number of descending plane partitions whose parts do not exceed n; also the number of n X n alternating sign matrices (ASM’s).