  We consider a triangular gap of side two in a $90^\circ$ angle on the
triangular lattice with mixed boundary conditions: a constrained, zig-zag
boundary along one side, and a free lattice line boundary along the other. We
study the interaction of the gap with the corner as the rest of the angle is
completely filled with lozenges. We show that the resulting correlation is
governed by the product of the distances between the gap and its three images
in the sides of the angle. The image in the side with constrained boundary has
the same orientation as the original gap, while the image in the side with free
boundary has the opposite orientation. This, together with the parallel between
the correlation of gaps in dimer packings and electrostatics we developed in
previous work, provides evidence for a unified way of understanding the
interaction of gaps with the boundary under mixed boundary conditions, which we
phrase as a conjecture. While the electrostatic interpretation is equivalent to
a steady state heat flow interpretation in the bulk, it turns out that the
latter view is more natural in the context of the interaction of the gaps with
the boundary.
  The starting point for our analysis is an exact formula we prove for the
number of lozenge tilings of certain trapezoidal regions with mixed boundary
conditions, which is equivalent to a new, multi-parameter generalization of a
classical plane partition enumeration problem (that of enumerating symmetric,
self-complementary plane partitions).
