aequationes mathematicae

, Volume 33, Issue 2–3, pp 230–250

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

• George E. Andrews
Research Papers

## Abstract

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

## Preview

### References

1. [1]
Andrews, G E,Applications of basic hypergeometric functions SIAM Rev16 (1974), 441–484Google Scholar
2. [2]
Andrews, G E,Plane partitions (III)the weak Macdonald conjecture Invent Math53 (1979), 193–225Google Scholar
3. [3]
Bailey, W N,Generalized Hypergeometric Series Cambridge University Press, London and New York, 1935 (Reprinted Hafner, New York, 1964)Google Scholar
4. [4]
Mills, W H, Robbins, D P, andRumsey Jr, H,Proof of the Macdonald conjecture Invent Math66 (1982), 73–87Google Scholar
5. [5]
Mills, W H, Robbins, D P, andRumsey, Jr, H,Enumeration of a symmetry class of plane partitions Working Paper No 822, April 1985, IDA, PrincetonGoogle Scholar
6. [6]
Sears, D B,On the transformation theory of hypergeometric functions Proc London Math Soc (2),52 (1950), 14–35Google Scholar
7. [7]
Stanley, R P,A baker's dozen of conjectures concerning plane partitions InCombinatoire énumérative (G Labelle and P Lerous eds), Lecture Notes in Math 1234 Springer, Berlin—New York, 1986, pp 285–293Google Scholar
8. [8]
Whittaker, E T andWatson G N,A Course of Modern Analysis 4th ed, Cambridge University Press, London and New York, 1927Google Scholar
9. [9]
Wilson, J A,Three-term contiguous relations and some new orthogonal polynomials InPade and Rational Approximation (E B Saff and R S Varga, Eds), Academic Press, New York, 1977, pp 227–232Google Scholar