  Gog and Magog trapezoids are certain arrays of positive integers that
generalize alternating sign matrices (ASMs) and totally symmetric
self-complementary plane partitions (TSSCPPs) respectively. Zeilberger used
constant term formulas to prove that there is the same number of (n,k)-Gog
trapezoids as there is of (n,k)-Magog trapezoids, thereby providing so far the
only proof for a weak version of a conjecture by Mills, Robbins and Rumsey from
1986. About 20 years ago, Krattenthaler generalized Gog and Magog trapezoids
and formulated an extension of their conjecture, and, recently, Biane and
Cheballah generalized Gog trapezoids further and formulated a related
conjecture. In this paper, we derive constant term formulas for various refined
enumerations of generalized Gog trapezoids including those considered by
Krattenthaler and by Biane and Cheballah. For this purpose we employ a result
on the enumeration of truncated monotone triangles which is in turn based in
the author's operator formula for the number of monotone triangles with
prescribed bottom row. As a byproduct, we also generalize the operator formula
for monotone triangles by including the inversion number and the complementary
inversion number for ASMs. Constant term formulas as well as determinant
formulas for the refined Magog trapezoid numbers that appear in Krattenthaler's
conjecture are also deduced by using the classical approach based on
non-intersecting lattice paths and the Lindstr\"om-Gessel-Viennot theorem.
Finally, we review and partly extend a few existing tools that may be helpful
in relating constant term formulas for Gogs to those for Magogs to eventually
prove the above mentioned conjectures.
