Fillings of skew shapes avoiding diagonal patterns

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