  Inspired by Propp's intruded Aztec diamond regions, we consider halved
hexagons in which two aligned arrays of triangular holes have been removed from
their boundaries. Unlike the intruded Aztec diamonds (whose numbers of domino
tilings contain some large prime factors in their factorizations), the numbers
of lozenge tilings of our doubly-intruded halved hexagons are given by simple
product formulas in which all factors are linear in the parameters. In this
paper, we present an extensive list of exact tiling enumerations of sixteen
different types of doubly-intruded halved hexagons. We also prove that the
lozenge tilings of a symmetric hexagon with three arrays of triangles removed
are always enumerated by a closed-form product formula. Our results generalize
several previous works, including Proctor's enumeration of the
transposed-complementary plane partitions, related work of Ciucu, and recent
generalizations of Rohatgi and of the author.
