### 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 Z_{n}(2, m) as a determinant {Mathematical expression} Conceivably this new representation may provide new interpretations of the combinatorial significance of 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

Original language | English (US) |
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Pages (from-to) | 230-250 |

Number of pages | 21 |

Journal | Aequationes Mathematicae |

Volume | 33 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 1987 |

### All Science Journal Classification (ASJC) codes

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