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

Abstract

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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Correspondence to Anna de Mier.

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de Mier, A. k-noncrossing and k-nonnesting graphs and fillings of ferrers diagrams. Combinatorica 27, 699–720 (2007). https://doi.org/10.1007/s00493-007-2297-2

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Mathematics Subject Classification (2000)

  • 05A19
  • 05A17
  • 05E10