|Stembridge's Theorem and Related Topics|
Alternating sign matrices (ASMs) and various plane partition
problems have been found to be closely related. A full account
of this relationship can be found in David Bressoud's book,
"Proofs and Confirmations, The Story of the Alternating Sign
A wonderfully engaging survey was prepared by David Robbins, "The Story of 1, 2, 7, 42, 429, 7436, ..." (The Math. Intelligencer, 13(1991), 12-16).
In 1986, Richard Stanley listed 13 plane partitions conjectures, including the TSSCPP (totally symmetric self-complementary plane partitions) conjecture (proved in J. Combin. Th., Ser.A, 66(1994), 28-39), which turned out to be instrumental in the Zeilberger proof of the original ASM conjecture. A related conjecture, the TSPP conjecture (totally symmetric plane partitions) was proved by John Stembridge using ingenious combinatorial insights, together with unparalleled expertise in Pfaffians.
In this talk, we shall begin with the history of my involvement in this topic. I will present an evaluation of the Vandermonde determinant that will illustrate the method used in my approach to these problems. Then we shall examine an alternative and much more heavy-handed approach to Stembridge's theorem. It will be completely parallel to the original proof of the TSSCPP conjecture.