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Gog and Magog trapezoids are certain arrays of positive integers thatgeneralize alternating sign matrices (ASMs) and totally symmetricself-complementary plane partitions (TSSCPPs) respectively. Zeilberger usedconstant term formulas to prove that there is the same number of (n,k)-Gogtrapezoids as there is of (n,k)-Magog trapezoids, thereby providing so far theonly proof for a weak version of a conjecture by Mills, Robbins and Rumsey from1986. About 20 years ago, Krattenthaler generalized Gog and Magog trapezoidsand formulated an extension of their conjecture, and, recently, Biane andCheballah generalized Gog trapezoids further and formulated a relatedconjecture. In this paper, we derive constant term formulas for various refinedenumerations of generalized Gog trapezoids including those considered byKrattenthaler and by Biane and Cheballah. For this purpose we employ a resulton the enumeration of truncated monotone triangles which is in turn based inthe author's operator formula for the number of monotone triangles withprescribed bottom row. As a byproduct, we also generalize the operator formulafor monotone triangles by including the inversion number and the complementaryinversion number for ASMs. Constant term formulas as well as determinantformulas for the refined Magog trapezoid numbers that appear in Krattenthaler'sconjecture are also deduced by using the classical approach based onnon-intersecting lattice paths and the Lindstr\"om-Gessel-Viennot theorem.Finally, we review and partly extend a few existing tools that may be helpfulin relating constant term formulas for Gogs to those for Magogs to eventuallyprove the above mentioned conjectures.

Constant term formulas for refined enumerations of Gog and Magog trapezoids