This material has been published in
Electron. J. Combin. 4(1) (1997), #R27, 62 pp.
Determinant identities and a generalization of the number of
totally symmetric self-complementary plane partitions
We prove a constant term conjecture of Robbins and Zeilberger
(J. Combin. Theory Ser. Ser. A 66 (1994), 17-27),
by translating the problem into a determinant evaluation problem and
evaluating the determinant. This determinant generalizes the
determinant that gives the number of all totally symmetric
self-complementary plane partitions contained in a
(2n)x(2n)x(2n) box and that was used by Andrews
(J. Combin. Theory Ser. A 66 (1994), 28-39) and Andrews and
Burge (Pacific J. Math. 158 (1993), 1-14) to compute this
number explicitly. The evaluation of the generalized determinant is
independent of Andrews and Burge's computations, and therefore in
particular constitutes a new solution to this famous enumeration
problem. We also evaluate a related determinant, thus
generalizing another determinant identity of Andrews and Burge (loc.
cit.). By translating some of our determinant identities into constant term
identities, we obtain several new constant term identities.
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