## Abstract

MacMahon's classical theorem on boxed plane partitions states that the generating function of the plane partitions fitting in an a×b×c box is equal to_{i=1}^{n}(1+q+q^{2}+…+q^{i−1}). By viewing a boxed plane partition as a lozenge tiling of a semi-regular hexagon, MacMahon's theorem yields a natural q-enumeration of lozenge tilings of the hexagon. However, such q-enumerations do not appear often in the domain of enumeration of lozenge tilings. In this paper, we consider a new q-enumeration of lozenge tilings of a hexagon with three bowtie-shaped regions removed from three non-consecutive sides. The unweighted version of the result generalizes a problem posed by James Propp on enumeration of lozenge tilings of a hexagon of side-lengths 2n, 2n+3, 2n, 2n+3, 2n, 2n+3 (in cyclic order) with the central unit triangles on the (2n+3)-sides removed. Moreover, our result also implies a q-enumeration of boxed plane partitions with certain constraints.

Original language | English (US) |
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Pages (from-to) | 23-57 |

Number of pages | 35 |

Journal | Advances in Applied Mathematics |

Volume | 82 |

DOIs | |

State | Published - Jan 1 2017 |

### Bibliographical note

Funding Information:This research was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation (grant no. DMS-0931945 ).

## Keywords

- Graphical condensation
- Lozenge tilings
- Perfect matchings
- Plane partitions