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

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

2006-02-09

arxiv.org/abs/math/0602195v2

Mathematics

Combinatorics

19 pages, 3 figures, section 3 expanded, added references

Scientific paper

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 Klazar to k>2. Moreover, this correspondence reinforces the links recently discovered by Krattenthaler between fillings of diagrams and the results of Chen et al.

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