## Abstract

A plane partition of the integer n is a representation of n in the form n = Σi, i≥1 n_{ij} where the integers n_{ij} are nonnegative and n_{ij} ≧ n_{i, j+l}n_{ij} ≧ n_{i+1}_{j}. In 1898, MacMahon conjectured that the generating function for the number of symmetric plane partitions (i.e., n_{ij} = n_{ji}) with each part at most m (i.e., n_{11}≦m) and at most s rows (i.e., n_{ij} = 0 for i> s) has a simple closed form. In 1972, Bender and Knuth conjectured that a simple closed form also exists for the generating function for plane partitions having at most s rows, n_{11} ≦ m and strict decrease along rows (i.e., n_{ij} > n_{i, j+1} whenever n_{ij} > 0). The main theorem of this paper establishes that each conjecture follows immediately from the other. In the first paper of this series, MacMahon’s conjecture was proved. Hence a corollary of the main theorem here is the truth of the Bender-Knuth conjecture; the Bender-Knuth conjecture has also been proved in a different manner by Basil Gordon.

Original language | English (US) |
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Pages (from-to) | 283-291 |

Number of pages | 9 |

Journal | Pacific Journal of Mathematics |

Volume | 72 |

Issue number | 2 |

DOIs | |

State | Published - Oct 1977 |

## All Science Journal Classification (ASJC) codes

- General Mathematics