# 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*Z*_{ n }(*x, m*) The special case*Z*_{ n }(1,*m*) is the generating function that arose in the weak Macdonald conjecture Mills—Robbins—Rumsey conjectured that*Z*_{ 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 of*Z*_{ n }(1,*m*) given previously Many results for the_{3}*F*_{2} 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 Conceivably this new representation may provide new interpretations of the combinatorial significance of

*Z*_{ n }(2,*m*) as a determinant$$Z_n (2,m) = \det \left( {\delta _{ij} + \sum\limits_{i = 0}^l {\left( {\begin{array}{*{20}c} {m + j + t} \\ t \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {m + i} \\ {m + t} \\ \end{array} } \right)} } \right)_{0 \leqq ij \leqq n - 1} $$

*Z*_{ n }(2,*m*) In the final analysis, one would like a combinatorial explanation of*Z*_{ n }(2,*m*) that would provide an algorithmic proof of the Mills Robbins—Rumsey conjecture### AMS (1980) subject classification

Primary 05A19, 10A45, 33A30## Preview

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### References

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## Copyright information

© Birkhauser Verlag 1987