  It is well-known that plane partitions, lozenge tilings of a hexagon, perfect
matchings on a honeycomb graph, and families of non-intersecting lattice paths
in a hexagon are all in bijection. In this work we consider regions that are
more general than hexagons. They are obtained by further removing
upward-pointing triangles. We call the resulting shapes triangular regions. We
establish signed versions of the latter three bijections for triangular
regions. We first investigate the tileability of triangular regions by
lozenges. Then we use perfect matchings and families of non-intersecting
lattice paths to define two signs of a lozenge tiling. Using a new method that
we call resolution of a puncture, we show that the two signs are in fact
equivalent. As a consequence, we obtain the equality of determinants, up to
sign, that enumerate signed perfect matchings and signed families of lattice
paths of a triangular region, respectively. We also describe triangular
regions, for which the signed enumerations agree with the unsigned
enumerations.
