### 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) |
---|---|

Pages (from-to) | 230-250 |

Number of pages | 21 |

Journal | Aequationes Mathematicae |

Volume | 33 |

Issue number | 2-3 |

DOIs | |

State | Published - Jun 1 1987 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

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*Aequationes Mathematicae*, vol. 33, no. 2-3, pp. 230-250. https://doi.org/10.1007/BF01840499

**Plane partitions IV : A conjecture of Mills-Robbins-Rumsey.** / Andrews, George E.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Plane partitions IV

T2 - A conjecture of Mills-Robbins-Rumsey

AU - Andrews, George E.

PY - 1987/6/1

Y1 - 1987/6/1

N2 - In this paper we consider a problem posed by W Mills D Robbins and H Rumsey for a certain plane partition generating function Zn(x, m) The special case Zn(1, m) is the generating function that arose in the weak Macdonald conjecture Mills-Robbins-Rumsey conjectured that Zn(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 Zn(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 of Zn(2, m) as a determinant {Mathematical expression} Conceivably this new representation may provide new interpretations of the combinatorial significance of Zn(2, m) In the final analysis, one would like a combinatorial explanation of Zn(2, m) that would provide an algorithmic proof of the Mills Robbins-Rumsey conjecture

AB - In this paper we consider a problem posed by W Mills D Robbins and H Rumsey for a certain plane partition generating function Zn(x, m) The special case Zn(1, m) is the generating function that arose in the weak Macdonald conjecture Mills-Robbins-Rumsey conjectured that Zn(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 Zn(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 of Zn(2, m) as a determinant {Mathematical expression} Conceivably this new representation may provide new interpretations of the combinatorial significance of Zn(2, m) In the final analysis, one would like a combinatorial explanation of Zn(2, m) that would provide an algorithmic proof of the Mills Robbins-Rumsey conjecture

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U2 - 10.1007/BF01840499

DO - 10.1007/BF01840499

M3 - Article

AN - SCOPUS:34250107722

VL - 33

SP - 230

EP - 250

JO - Aequationes Mathematicae

JF - Aequationes Mathematicae

SN - 0001-9054

IS - 2-3

ER -