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