On refined enumerations of totally symmetric self-complementary plane partitions II

In this paper we settle a weak version of a conjecture (i.e. Conjecture 6) by Mills, Robbins and Rumsey in the paper "Self-complementary totally symmetric plane partitions" (J. Combin. Theory Ser. A 42, 277-292). In other words we show that the number of shifted plane partitions invariant under an involution is equal to the number of alternating sign matrices invariant under the vertical flip. We also give a determinantal formula for the general conjecture (Conjecture 6), but this determinant is still hard to evaluate. In this paper we introduce two new classes of domino plane partitions, one has the same cardinality as the set of half-turn symmetric alternating sign matrices and the other has the same cardinality as the set of vertically symmetric alternating sign matrices.