  In the paper [J. Combin. Theory Ser. A 43 (1986), 103--113], Stanley gives
formulas for the number of plane partitions in each of ten symmetry classes.
This paper together with results by Andrews [J. Combin. Theory Ser. A 66
(1994), 28-39] and Stembridge [Adv. Math 111 (1995), 227-243] completes the
project of proving all ten formulas.
  We enumerate cyclically symmetric, self-complementary plane partitions. We
first convert plane partitions to tilings of a hexagon in the plane by
rhombuses, or equivalently to matchings in a certain planar graph. We can then
use the permanent-determinant method or a variant, the Hafnian-Pfaffian method,
to obtain the answer as the determinant or Pfaffian of a matrix in each of the
ten cases. We row-reduce the resulting matrix in the case under consideration
to prove the formula. A similar row-reduction process can be carried out in
many of the other cases, and we analyze three other symmetry classes of plane
partitions for comparison.
