k-noncrossing and k-nonnesting graphs and fillings of Ferrers diagrams

We study the distribution of k-crossings and k-nestings and other patterns in ordered graphs by using fillings of Ferrers diagrams. The main result states that there are as many graphs without k-crossings as without k-nestings. We also show that studying equirrestrictive patterns in ordered graphs is equivalent to studying equirrestrictive matrices in fillings of Ferrers diagrams.