Crossings and nestings of matchings and partitions

@article{Chen2005CrossingsAN,
  title={Crossings and nestings of matchings and partitions},
  author={William Y. C. Chen and Eva Yu-Ping Deng and Rosena R. X. Du and Richard P. Stanley and Catherine Huafei Yan},
  journal={Transactions of the American Mathematical Society},
  year={2005},
  volume={359},
  pages={1555-1575}
}
We present results on the enumeration of crossings and nestings for matchings and set partitions. Using a bijection between partitions and vacillating tableaux, we show that if we fix the sets of minimal block elements and maximal block elements, the crossing number and the nesting number of partitions have a symmetric joint distribution. It follows that the crossing numbers and the nesting numbers are distributed symmetrically over all partitions of [n], as well as over all matchings on [2n… Expand

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