Plane partitions IV: A conjecture of Mills-Robbins-Rumsey

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Abstract

In this paper we consider a problem posed by W Mills D Robbins and H Rumsey for a certain plane partition generating function Zn(x, m) The special case Zn(1, m) is the generating function that arose in the weak Macdonald conjecture Mills-Robbins-Rumsey conjectured that Zn(2, m) also possesses a nice finite product representation Their conjecture is proved as Theorem 1 The method of proof resembles that of the evaluation of Zn(1, m) given previously Many results for the3F2 hypergeometric function are required including Whipple's theorem, the Pfaff-Saalschutz summation and contiguous relations In passing we note that our Lemma 2 provides a new and simpler representation of Zn(2, m) as a determinant {Mathematical expression} Conceivably this new representation may provide new interpretations of the combinatorial significance of Zn(2, m) In the final analysis, one would like a combinatorial explanation of Zn(2, m) that would provide an algorithmic proof of the Mills Robbins-Rumsey conjecture

Original languageEnglish (US)
Pages (from-to)230-250
Number of pages21
JournalAequationes Mathematicae
Volume33
Issue number2-3
DOIs
StatePublished - Jun 1 1987

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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