aequationes mathematicae

, Volume 33, Issue 2–3, pp 230–250 | Cite as

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

  • George E. Andrews
Research Papers


In this paper we consider a problem posed by W Mills D Robbins and H Rumsey for a certain plane partition generating functionZ n (x, m) The special caseZ n (1,m) is the generating function that arose in the weak Macdonald conjecture Mills—Robbins—Rumsey conjectured thatZ n (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 ofZ n (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 ofZ n (2,m) as a determinant
$$Z_n (2,m) = \det \left( {\delta _{ij} + \sum\limits_{i = 0}^l {\left( {\begin{array}{*{20}c} {m + j + t} \\ t \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {m + i} \\ {m + t} \\ \end{array} } \right)} } \right)_{0 \leqq ij \leqq n - 1} $$
Conceivably this new representation may provide new interpretations of the combinatorial significance ofZ n (2,m) In the final analysis, one would like a combinatorial explanation ofZ n (2,m) that would provide an algorithmic proof of the Mills Robbins—Rumsey conjecture

AMS (1980) subject classification

Primary 05A19, 10A45, 33A30 


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Copyright information

© Birkhauser Verlag 1987

Authors and Affiliations

  • George E. Andrews
    • 1
  1. 1.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

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