D. Zeilberger, Proof of the alternating sign matrix conjecture, Electr. J. Combin. 7 (2000) R37; G. Kuperberg, Another proof of the alternating sign matrix conjecture, Int. Math. Res. Notes (1996) 139-150, math.CO/9712207.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....2. 7 (Kuperberg) The ( 1) enumeration of cyclically symmetric self complementary plane partitions in a box with sides 2 2 2 with weight ( 1) is the square root of the ordinary enumeration, that is (3k 1) k) 22) This is also the number of alternating sign matrices (see [37]) the number of totally symmetric self complementary plane partitions [4] and the number of descending plane partitions. Results for small values of suggest that the sign is 1 for all . Thus, the only cases that are still open are the case of symmetric transpose complementary plane ....

D. Zeilberger, Proof of the alternating sign matrix conjecture, Electron. J. Combin. 3(2) \The Foata Festschrift" (1996.

....antichains in this poset. For other connections, see [La] and for more information about the Ehresmann Bruhat order on finite Coxeter groups, see [GK] The problem of enumerating alternating sign matrices, due to Mills, Robbins, and Rumsey [MRR1] MRR2] has resisted many attempts, till Zeilberger [Z1] gave a solution with the collaboration of his computer, the control of eighty eight of his friends (he has a lot more) and many references to the Bible. Shorter proofs were given by Kuperberg [Ku] and [Z2] again, after realizing that the partition function of square ice models had already been ....

D. Zeilberger. Proof of the alternating sign matrix conjecture. Electronic J. Comb. 3 (1996) R 13.

....there is a bijection[1] between alternating sign matrices (ASMs) and n certain osculating paths. An example of this bijection is illustrated in figure 2. The expression of RRM[2, 3] a n = n 1 i=1 (3i 1) n i) 1) for the number of nn ASMs was first impressively proved by Zeilberger [4] who related it to a particular class of plane partitions. These partitions had been enumerated by Andrews [5] based on a result by Stembridge. A shorter derivation was subsequently obtained by Kuperberg[6] using a result of Izergin and Korepin[7] A pure combinatorial Figure 1: An example of ....

Doron Zeilberger. Proof of the alternating sign matrix conjecture. Elec. J. of Combinatorics, 3, 1996.

.... Numerous conjectures concerning ASMs were put forward by Mills, Robbins, and Rumsey in [7] These are further described in [11] A general formula for the number of n n ASMs was conjectured by Mills Robbins Rumsey to be n 1 # k=0 (3k 1) n k) 33) and proved by Zeilberger in 1996 [13]. Shorter proofs were subsequently given by Kuperberg [5] and a refinement by Zeilberger [14] There is a substantial amount of combinatorics concerning ASMs that is still not fully understood. Proofs for the number of ASMs having symmetries (e.g. invariant under reflection about a vertical axis, ....

D. Zeilberger. "Proof of the Alternating Sign Matrix Conjecture." Electronic J. Combinatorics 3, No. 2 (1996), R13, pp. 1--84.

.... by David Bressoud [16] The alternating sign matrix conjecture concerns the total number of n n alternating sign matrices, which was conjectured by Mills, Robbins, and Rumsey to be Q n 1 j=0 (3j 1) n j) The problem was open for fteen years until it was nally settled by Zeilberger [80]. The development of ideas is described in the book by Bressoud. There are deep relations to various parts of Algebraic Combinatorics, especially to plane partitions, where the same counting function occurred, and also to Statistical Mechanics, where the con guration of water molecules in square ....

D. Zeilberger, \Proof of the alternating sign matrix conjecture", Electron. J. Combin. 3, 1996 #R13, also in: The Foata Festschrift, ed. J. Desarmemien, A. Kerber, and V. Strehl, 289 - 273, Gap, France.

.... by David Bressoud [16] The alternating sign matrix conjecture concerns the total number of n n alternating sign matrices, which was conjectured by Mills, Robbins, and Rumsey to be # n 1 j=0 (3j 1) n j) The problem was open for fifteen years until it was finally settled by Zeilberger [80]. The development of ideas is described in the book by Bressoud. There are deep relations to various parts of Algebraic Combinatorics, especially to plane partitions, where the same counting function occurred, and also to Statistical Mechanics, where the configuration of water molecules in square ....

D. Zeilberger, "Proof of the alternating sign matrix conjecture", Electron. J. Combin. 3, 1996 #R13, also in: The Foata Festschrift,ed.J.Desarmemien, A. Kerber, and V. Strehl, 289 -- 273, Gap, France.

....of such partitions is given by T (n) n 1 # j=0 (3j 1) n j) 1) 1 Current address: Department of Mathematics, Eberly College of Science, The Pennsylvania State University, University Park, PA 16802 1 (This formula also gives the number of n n alternating sign matrices. See [9]. The values T (n) can be found as sequence A005130 in [8] Another family mentioned by Bressoud is the set of cyclically symmetric transpose complement plane partitions (CSTCPPs) We will let C(n) denote the number of such partitions that fit in a 2n2n2n box. The values C(n) make up sequence ....

D. Zeilberger, Proof of the alternating sign matrix conjecture, Electronic Journal of Combinatorics 3 (1996), R13. http://www.research.att.com/#njas/sequences/

....understanding of the role of planarity in the theory. 2. Alternating sign patterns A determinant formula found by the physicists Korepin and Izergin provided the key to Greg Kuperberg s elegant solution [K5] to the alternatingsign matrix problem (solved by Doron Zeilberger several years earlier [Z2]) This formula describes the behavior of the six vertex model on an n by n square lattice with speci c boundary conditions (arrows pointing in along the left and right, arrows pointing out along the top and bottom) Now another 5 physicist, Osamu Tsuchiya, has discovered a di erent determinant ....

D. Zeilberger, Proof of the alternating sign matrix conjecture, The Foata Festschrift, Electron. J. Combin. 3 (1996), Research Paper 13.

.... : Numerous conjectures concerning ASMs were put forward by Mills, Robbins, and Rumsey in [7] These are further described in [11] A general formula for the number of n n ASMs was conjectured by Mills Robbins Rumsey to be n 1 Y k=0 (3k 1) n k) 33) and proved by Zeilberger in 1996 [13]. Shorter proofs were subsequently given by Kuperberg [5] and a re nement by Zeilberger [14] There is a substantial amount of combinatorics concerning ASMs that is still not fully understood. Proofs for the number of ASMs having symmetries (e.g. invariant under re ection about a vertical axis, ....

D. Zeilberger. \Proof of the Alternating Sign Matrix Conjecture." Electronic J. Combinatorics 3, No. 2 (1996), R13, pp. 1-84.

.... n, for small values of n, goes like 1;2;7;42;429;7436; and it was conjectured by Mills et al. that the number of ASMs of order n is given by the product 1 4 7 (3n 2) n (n 1) n 2) 2n 1) However, it took over a decade before this conjecture was proved by Zeilberger [30]. For more details on this history, see the expository article by Robbins [21] the survey article by Bressoud and Propp [7] or the book by Bressoud [6] Here my concern will be not with the alternating sign matrix conjecture and its proof by Zeilberger, but with the inherent interest of ....

....partial sums, the ith row has i 1 s in it and n i 0 s. Hence we may form a triangular array whose ith row consists of precisely those values j for which the i; jth entry of the partial sum matrix is 1. The result is called a monotone triangle [17] or Gog triangle in the terminology of Zeilberger [30]) Figure 7 shows the seven monotone triangles of order 3. 4 James Propp 0 B B 0 1 0 0 1 1 1 0 0 0 0 1 0 1 0 0 1 C C A 0 B B 0 1 0 0 1 0 1 0 1 0 1 1 1 1 1 1 1 C C A 2 1 3 1 3 4 1 2 3 4 Figure 6: Turning an ASM into a monotone triangle. 3 2 3 1 2 3 3 1 3 1 2 3 2 2 3 1 2 3 2 ....

[Article contains additional citation context not shown here]

D. Zeilberger, Proof of the alternating sign matrix conjecture, Electronic J. Comb. 3 (1996), R13; arXiv:math.CO/9407211.

....problem of determining the number of alternating sign matrices of a given order turned out to be among the hardest in enumerative combinatorics. In 1983, Mills, Robbins and Rumsey [MRR] conjectured that this number is given by a certain simple product formula. This was first proved by Zeilberger [Z] and later by Kuperberg [Ku1] Let ASM (n) be the set of alternating sign matrices of order n. Weight each A 2 ASM (n) by x N Gamma (A) and let An (x) be their generating function. It turns out that for x = 2 we can determine various refinements of A(x) using the Reduction Theorem. Indeed, ....

D. Zeilberger, Proof of the alternating-sign matrix conjecture, Electronic J. Combin. , to appear.

....there is a bijection[1] between alternating sign matrices (ASMs) and n certain osculating paths. An example of this bijection is illustrated in figure 2. The expression of RRM[2, 3] a n = n 1 Y i=1 (3i 1) n i) 1) for the number of n n ASMs was first impressively proved by Zeilberger [4] who related it to a particular class of plane partitions. These partitions had been enumerated by Andrews [5] based on a result by Stembridge. A shorter derivation was subsequently obtained by Kuperberg[6] using a result of Izergin and Korepin[7] A pure combinatorial Figure 1: An example of ....

Doron Zeilberger. Proof of the alternating sign matrix conjecture. Elec. J. of Combinatorics, 3, 1996.

....way. So in order to find out whether object A equals object B, all we have to do is find their canonical forms, c(A)andc(B) and check whether or not c(A)equalsc(B) Example 1.2.1. Prove the following identity The Third Author of This Book = The Prover of the Alternating Sign Matrix Conjecture [Zeil95a]. Solution: First verify that both sides of the identity are objects that belong to a well defined class that possesses a canonical form. In this case the class is that of citizens of the USA, and a good canonical form is the Social Security number. Next compute (or look up) the Social Security ....

Zeilberger, Doron, Proof of the alternating sign matrix conjecture, Electronic J. Combinatorics, to appear. [Also available by anon. ftp to ftp.math.temple.edu, in the file /pub/zeilberg/asm/asm.ps or on the WWW at http://www.math.temple.edu/#zeilberg.]

No context found.

D. Zeilberger, Proof of the alternating sign matrix conjecture, Electr. J. Combin. 7 (2000) R37; G. Kuperberg, Another proof of the alternating sign matrix conjecture, Int. Math. Res. Notes (1996) 139-150, math.CO/9712207.

No context found.

D. Zeilberger, "Proof of the alternating sign matrix conjecture", The Electronic Journal of Combinatorics 3, 1996 #R13, also in: The Foata Festschrift, ed. J. Desarmemien, A. Kerber, and V. Strehl, 289 -- 273, Gap, France.

No context found.

D. Zeilberger, Proof of the alternating sign matrix conjecture, Electr. J. Combin. 7 (2000) R37; G. Kuperberg, Another proof of the alternating sign matrix conjecture, Int. Math. Res. Notes (1996) 139-150, math.CO/9712207.

No context found.

D. Zeilberger, "Proof of the alternating sign matrix conjecture", Electron. J. Combin. 3, 1996 #R13, also in: The Foata Festschrift, ed. J. Desarmemien, A. Kerber, and V. Strehl, 289 -- 273, Gap, France.

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