aequationes mathematicae

, Volume 33, Issue 1, pp 230–250 | Cite as

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
$$Z_n (2,m) = \det \left( {\delta _{ij} + \sum\limits_{t = 0}^1 {\left( {\mathop {m + j + t}\limits_t } \right)} \left( {\mathop {m + t}\limits_{m + t} } \right)} \right)_{0 \leqq ij \leqq n - 1} $$
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

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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

Copyright information

© Birkhauser Verlag 1987

Authors and Affiliations

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

Personalised recommendations