Three product formulas for ratios of tiling counts of hexagons with collinear holes

Rosengren found an explicit formula for a certain weighted enumeration of lozenge tilings of a hexagon with an arbitrary triangular hole. He pointed out that a certain ratio corresponding to two such regions has a nice product formula. In this paper, we generalize this to hexagons with arbitrary collinear holes. It turns out that, by using same approach, we can also generalize Ciucu's work on the number and the number of centrally symmetric tilings of a hexagon with a fern removed from its center. This proves a recent conjecture of Ciucu.