Symmetries of plane partitions and the permanent—determinant method

Abstract

R. P. Stanley (1986, J. Combin. Theory Ser. A 43, 103–113) gives formulas for the number of plane partitions in each of 10 symmetry classes. This paper together with papers of G. Andrews ( J. Combin. Theory Ser. A, to appear) and J. Stembridge (The enumeration of totally symmetric plane partitions, preprint) completes the project of proving all 10 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 10 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.