  Totally symmetric self-complementary plane partitions (TSSCPPs) are boxed
plane partitions with the maximum possible symmetry. We use the well-known
representation of TSSCPPs as a dimer model on a honeycomb graph enclosed in
one-twelfth of a hexagon with free boundary to express them as perfect
matchings of a family of non-bipartite planar graphs. Our main result is that
the edges of the TSSCPPs form a Pfaffian point process, for which we give
explicit formulas for the inverse Kasteleyn matrix. Preliminary analysis of
these correlations are then used to give a precise conjecture for the limit
shape of TSSCPPs in the scaling limit.
