On the Symmetry of the Distribution of k-Crossings and k-Nestings in Graphs

This note contains two results on the distribution of k-crossings and k-nestings in graphs. On the positive side, we exhibit a class of graphs for which there are as many k-noncrossing 2-nonnesting graphs as k-nonnesting 2-noncrossing graphs. This class consists of the graphs on [n] where each vertex x is joined to at most one vertex y with y < x. On the negative side, we show that this is not the case if we consider arbitrary graphs. The counterexample is given in terms of fillings of Ferrers diagrams and solves a problem of Krattenthaler.