We generalize a special case of a theorem of Proctor on the enumeration of lozenge tilings of a hexagon with a maximal staircase removed, using Kuo's graphical condensation method. Additionally, we prove a formula for a weighted version of the given region. The result also extends work of Ciucu and Fischer. By applying the factorization theorem of Ciucu, we are also able to generalize a special case of MacMahon's boxed plane partition formula.

The work of Mills, Robbins, and Rumsey on cyclically symmetric plane partitions yields a simple product formula for the number of lozenge tilings of a regular hexagon, which are invariant under roation by 120 • . In this paper we generalize this result by enumerating the cyclically symmetric lozenge tilings of a hexagon in which four triangles have been removed in the center.

We prove exact product formulas for the tiling generating functions of various halved hexagons and quartered hexagons with defects on boundary. Our results generalize the previous work of the first author and the work of Ciucu.

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