  A skew shape is the difference of two top-left justified Ferrers shapes
sharing the same top-left corner. We study integer fillings of skew shapes. As
our first main result, we show that for a specific hereditary class of skew
shapes, which we call D-free shapes, the fillings that avoid a north-east chain
of size $k$ are in bijection with fillings that avoid a south-east chain of the
same size. Since Ferrers shapes are a subclass of D-free shapes, this result
can be seen as a generalization of previous analogous results for Ferrers
shapes.
  As our second main result, we construct a bijection between 01-fillings of an
arbitrary skew shape that avoid a south-east chain of size 2, and the
01-fillings of the same shape that simultaneously avoid a north-east chain of
size 2 and a particular non-square subfilling. This generalizes a previous
result for transversal fillings.
