A dual of MacMahon's theorem on plane partitions

Abstract

A classical theorem of MacMahon states that the number of lozenge tilings of any centrally symmetric hexagon drawn on the triangular lattice is given by a beautifully simple product formula. In this paper we present a counterpart of this formula, corresponding to the {\it exterior} of a concave hexagon obtained by turning 120 degrees after drawing each side (MacMahon's hexagon is obtained by turning 60 degrees after each step).

22 pages

References in corpus (3)

• Proof of George Andrews's and David Robbins's q-TSPP Conjecture
• A triangular gap of size two in a sea of dimers on a $60^{\circ }$ angle
• The interaction of collinear gaps of arbitrary charge in a two dimensional dimer system

Cited by in corpus (15)

• A shuffling theorem for lozenge tilings of doubly-dented hexagons
• Tiling enumeration of doubly-intruded halved hexagons
• Another dual of MacMahon's theorem on plane partitions
• A Shuffling Theorem for Reflectively Symmetric Tilings
• Plane partitions of shifted double staircase shape
• Lozenge tilings of hexagons with central holes and dents
• Tiling Enumeration of Hexagons with Off-central Holes
• Lozenge tilings of doubly-intruded hexagons
• Centrally symmetric tilings of fern-cored hexagons
• Inverting the Kasteleyn matrix for holey hexagons
• A generalization of Kuo condensation
• Enumeration of lozenge tilings of a hexagon with a shamrock missing on the symmetry axis
• Skew doubled shifted plane partitions: calculus and asymptotics
• Problems in the Enumeration of Tilings
• Enumeration of symmetric centered rhombus tilings of a hexagon