  We $q$-enumerate lozenge tilings of a hexagon with three bowtie-shaped
regions have been removed from three non-consecutive sides. The unweighted
version of the result generalizes a problem posed by James Propp on enumeration
of lozenge tilings of a hexagon of side-lengths $2n,2n+3,2n,2n+3,2n,2n+3$ (in
cyclic order) with the central unit triangles on the $(2n+3)$-sides removed.
