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The enumeration of lozenge tilings of hexagons with holes has received muchattention during the last three decades. One notable feature is that a lot ofthe recent development involved Kuo's graphical condensation. Motivated byCiucu, Lai and Rohatgi's work on tilings of hexagons with a removed triad ofbowties, in this paper, we show that the ratio of numbers of lozenge tilings oftwo more general regions is expressed as a simple product formula. Our proofdoes not involve the graphical condensation method. The proof is short anddirect. We also provide a corresponding formula for cyclically symmetriclozenge tilings. Several previous results can be easily deduced from ourgeneralization.

Lozenge tilings of hexagons with holes on three crossing lines