Lozenge tilings of hexagons with intrusions. I: Generalized intrusion (Q6632116)

MacMahon's theorem on boxed plane partitions gives an elegant formula for the number of lozenge tilings of hexagons with side lengths \(x\), \(y\), \(z\), \(x\), \(y\), \(z\) arranged in clockwise order. In this paper, the authors consider a new hexagonal region with intrusions, that is, a region obtained from hexagons by removing consecutive triangles of various sizes along the perpendicular bisector of the left side, such that the triangles alternate in orientation. A weight of \(\frac{q^i+q^{-1}}{2}\) is assigned to certain types of lozenges, and the authors prove that the tiling generating function (TGF) of these new regions is given by a product formula (Theorem 2.2). Their main approach employs \textit{E. H. Kuo}'s graphical condensation [Theor. Comput. Sci. 319, No. 1--3, 29--57 (2004; Zbl 1043.05099)] by expressing their TGF recursively in terms of TGFs of smaller regions from the same family or other well-studied regions; they can be obtained from \textit{M. Ciucu}'s matching factorization theorem [J. Comb. Theory, Ser. A 77, No. 1, 67--97 (1997; Zbl 0867.05055)] and the results of \textit{T. Lai} and \textit{R. Rohatgi} [Ann. Comb. 25, No. 2, 471--493 (2021; Zbl 1466.05030)].