| In the early 20th century, MacMahon proved that the number of lozenge tilings of a hexagon with side-lengths a,b,c,a,b,c in this cyclic order is given by a simple product formula. More recently, several mathematicians have proven formulas for the number of tilings of hexagons with some type of defects. The results here are in the vein of Proctor, and Ciucu and Fischer. We prove formulas for two types of hexagonal regions with a maximal staircase removed and one other boundary defect. Furthermore, we provide formulas for particular weighted counterparts of these two regions. Having both the weighted and unweighted results in hand, we apply Ciucu's factorization theorem to get formulas for the number of tilings of symmetric hexagons with either two or three boundary defects in specific positions. In addition, we enumerate the tilings of a hexagonal region with two staircases removed and a boundary defect, and provide a formula for the matching generating function of a weighted counterpart. |
