Lozenge Tilings of Hexagons with Intrusions I: Generalized Intrusion

MacMahon's classical theorem on the number of boxed plane partitions has been generalized in several directions. One way to generalize the theorem is to view boxed plane partitions as lozenge tilings of a hexagonal region and then generalize it by making some holes in the region and counting its tilings. In this paper, we provide new regions whose numbers of lozenges tilings are given by simple product formulas. The regions we consider can be obtained from hexagons by removing structures called intrusions. In fact, we show that the tiling generating functions of those regions under certain weights are given by similar formulas. These give the $q$-analogue of the enumeration results.