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The story of 1,2,7,42,429,7436,.. (English) Zbl 0723.05004

This is a fascinating and very readable account of how the numbers \(S_ n=\prod^{n-1}_{i=0}\frac{(3i+1)!}{(n+i)!},\) which have been shown by G. E. Andrews to count descending plane partitions, appear also to count both alternating sign matrices and self-complementary total symmetric plane partitions. Other conjectures relating to plane partitions and alternating sign matrices are given. The paper covers work due to Robbins, Mills, Rumsey, Andrews, Stanley and I. G. Macdonald. “These conjectures are of such compelling simplicity that it is hard to understand how any mathematician can bear the pain of living without understanding why they are true.”

MSC:

05A15 Exact enumeration problems, generating functions
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References:

[1] G. E. Andrews, Plane Partitions (III): The weak Mac-donald conjecture,Invent. Math. 53 (1979), 193–225. · Zbl 0421.10011 · doi:10.1007/BF01389763
[2] Guy David and Carlos Tomei, The problem of calissons,Amer. Math. Monthly 96 (1989), 429–431. · Zbl 0723.05037 · doi:10.2307/2325150
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[11] W. H. Mills, David P. Robbins, and Howard Rumsey, Jr., Enumeration of a symmetry class of plane partitions,Discrete Mathematics 67 (1987), 43–55. · Zbl 0656.05006 · doi:10.1016/0012-365X(87)90165-8
[12] W. H. Mills, David P. Robbins, and Howard Rumsey, Jr., Self-complementary totally symmetric plane partitions.J. Combin. Theory Ser. A 42 (1986), 277–292. · Zbl 0615.05011 · doi:10.1016/0097-3165(86)90098-1
[13] David P. Robbins and Howard Rumsey, Jr., Determinants and alternating sign matrices,Advances in Mathematics 62 (1986), 169–184. · Zbl 0611.15008 · doi:10.1016/0001-8708(86)90099-X
[14] Richard P. Stanley, Symmetries of plane partitions,J. Combin. Theory Ser. A 43 (1986), 103–113. Erratum, 44 (1987), 310. · Zbl 0602.05007 · doi:10.1016/0097-3165(86)90028-2
[15] Richard P. Stanley, A baker’s dozen of conjectures concerning plane partitions,Combinatoire Enumerative, Lecture Notes in Mathematics, Vol. 1234, New York: Springer-Verlag (1985), 285–293.
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