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MacMahon's theorem on plane partitions yields a simple product formula fortiling number of a hexagon, and Cohn, Larsen and Propp's theorem provides anexplicit enumeration for tilings of a dented semihexagon via semi-strictGelfand--Tsetlin patterns. In this paper, we prove a natural hybrid of the twotheorems for hexagons with an arbitrary set of unit triangles removed along ahorizontal axis. In particular, we show that the `shuffling' of removed unittriangles only changes the tiling number of the region by a simplemultiplicative factor. Our main result generalizes a number of knownenumerations and asymptotic enumerations of tilings. We also reveal connectionsof the main result to the study of symmetric functions and $q$-series.

A shuffling theorem for lozenge tilings of doubly-dented hexagons