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

Anna de Mier

Abstract

Given a graph on [n], two edges (i,j) and (k,l) are a crossing if i>k>j>l and they are a nesting if i<k<l<j. A k-crossing (k-nesting) is a set of k edges every two of them being a crossing (nesting). Our aim is to show that the number of graphs containing no k-crossing is the same as the number of graphs containing no k-nesting. To do so, we give a correspondence between graphs with a given degree sequence and fillings of Ferrers diagrams by nonnegative integers with prescribed row and column sums. In this setting, k-crossings and k-nestings of the graph become occurrences of the identity and the antiidentity matrices in the filling. We use this to show the equality of the numbers of k-noncrossing and k-nonnesting graphs with a given degree sequence. This generalizes the analogous result for matchings and partition graphs of Chen, Deng, Du, Stanley, and Yan, and extends results of