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In a recent preprint, Lai showed that the quotient of generating functions ofweighted lozenge tilings of two "half hexagons with lateral dents", whichdiffer only in width, factors nicely, and the same is true for the quotient ofgenerating functions of weighted lozenge tilings of two "quarter hexagons withlateral dents". Lai achieved this by using "graphical condensation" (i.e.,application of a certain Pfaffian identity to the weighted enumeration ofmatchings).  The purpose of this note is to exhibit how this can be done by theLindstr\"om--Gessel--Viennot method for nonintersecting lattice paths. For thecase of "half hexagons", basically the same observation, but restricted to mereenumeration (i.e., all weights of lozenge tilings are equal to $1$), iscontained in a recent preprint of Condon.

The quotient of generating functions of lozenge tilings for certain regions derived from hexagons, obtained with non--intersecting lattice paths