  We prove a constant term conjecture of Robbins and Zeilberger (J. Combin.
Theory 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 $(2n)\times(2n)\times(2n)$
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.

Combinatorics