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

A triangular gap of size two in a sea of dimers in a $90^\circ$ angle  with mixed boundary conditions, and a heat flow conjecture for the general  case