  We put recent results by Chen, Deng, Du, Stanley and Yan on crossings and
nestings of matchings and set partitions in the larger context of the
enumeration of fillings of Ferrers shape on which one imposes restrictions on
their increasing and decreasing chains. While Chen et al. work with
Robinson-Schensted-like insertion/deletion algorithms, we use the growth
diagram construction of Fomin to obtain our results. We extend the results by
Chen et al., which, in the language of fillings, are results about
$0$-$1$-fillings, to arbitrary fillings. Finally, we point out that, very
likely, these results are part of a bigger picture which also includes recent
results of Jonsson on $0$-$1$-fillings of stack polyominoes, and of results of
Backelin, West and Xin and of Bousquet-M\'elou and Steingr\'\i msson on the
enumeration of permutations and involutions with restricted patterns. In
particular, we show that our growth diagram bijections do in fact provide
alternative proofs of the results by Backelin, West and Xin and by
Bousquet-M\'elou and Steingr\'\i msson.
