Dept. of Mathematics, Tottori University, Koyama, Japan.

In this talk we give Pfaffian or determinant expressions and constant term identities for the conjectures in the paper “Self-complementary totally symmetric plane partitions” (J. Combin. Theory Ser. A 42, (1986), 277 – 292) by W.H. Mills, D.P. Robbins and H. Rumsey. These conjectures are about weighted enumerations of certain plane partitions and have been open more than 20 years. We also settle a weak version of Conjecture in the paper, i.e., the number of shifted plane partitions invariant under a certain involution is equal to the number of alternating sign matrices invariant under the vertical flip.