Proof of the Alternating Sign Matrix Conjecture

Abstract

Mills, Robbins, and Rumsey conjectured, and Zeilberger proved, that the number of alternating sign matrices of order n equals A(n) := 1!4!7!···(3n−2)! n!(n+1)!···(2n−1)! . Mills, Robbins, and Rumsey also made the stronger conjecture that the number of such matrices whose (unique) '1' of the first row is at the r th column, equals A(n) n+r−2 n−1 2n−1−r n−1

Advanced Determinant Calculus

“…Alternating sign matrix are currently the most fascinating, and most mysterious, objects in enumerative combinatorics. The reader is referred to [18,19,111,148,97,198,199] for more detailed material. Incidentally, the "birth" of alternating sign matrices came through -determinants, see [150].…”

“…12 The program is based on the observation that any "closed form" sequence (a n ) n≥0 that appears within the "hypergeometric paradigm" is either given by a rational expression, like a n = n/(n + 1), or the sequence of successive quotients (a n+1 /a n ) n≥0 is given by a rational expression, like in the case of central binomial coefficients a n = 2n n , or the sequence of successive quotients of successive quotients ((a n+2 /a n+1 )/(a n+1 /a n )) n≥0 is given by a rational expression, like in the case of the famous sequence of numbers of alternating sign matrices (cf. the paragraphs following (3.9), and [18,19,111,148,97,198,199] for information on alternating sign matrices), a n = n−1 i=0 (3i + 1)! (n + i)!…”

Arctic boundaries of the ice model on three-bundle domains

“…Instead of attempting to implement this in the fullest possible generality, for the sake of specificity we only focus on a particular example. Perhaps the simplest (non free-fermionic) case would be the domain-wall six-vertex model at ice point, where the weights of all six vertex types are equal; this model received considerable interest over the past three decades due to its relationship with alternating sign matrices [17,36,37,48,52,62,63]. In order to demonstrate the versatility of this framework, and since it will change little in the proof, we will in fact analyze a more general situation, given by the ice model on the three-bundle domain.…”

“…As explained earlier, our proofs are based on a justification of the tangent method, which requires access to asymptotics for the singly refined correlation function of the vertex model of interest. At ice point, these quantities can be expressed exactly due to the works [8] of Cantini-Sportiello on the three-bundle domain and [63] of Zeilberger in the domain-wall case.…”

Sign matrix polytopes from Young tableaux

“…Alternating sign matrices were introduced by Robbins and Rumsey in their study of the λ-determinant [22], with an enumeration formula conjectured by Mills, Robbins, and Rumsey [19]. The proof of this conjecture [31,18] was a major accomplishment in enumerative combinatorics in the 1990's. Alternating sign matrices are still a source of interest, in particular, with regard to intriguing open bijective questions involving plane partitions and connections to both the six-vertex model and various loop models in statistical physics [3,5,12,7,11,10,20,21,26,28,29,27].…”