The equivalence between enumerating cyclically symmetric, self-complementary and totally symmetric, self-complementary plane partitions

We prove that the number of cyclically symmetric, self-complementary plane partitions contained in a cube of side 2n equals the square of the number o f totally symmetric, self-complementary plane partitions contained in the s ame cube, without explicitly evaluating either of these numbers. This appea rs to be the first direct proof of this fact. The problem of finding such a proof was suggested by Stanley (J. Combin. Theory Ser. A 43 (1996)1 103-11 3). (C) 1999 Academic Press.