Crossings and Nestings of Matchings and Partitions
Abstract
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 $[2n]$. As a corollary, the number of $k$noncrossing partitions is equal to the number of $k$nonnesting partitions. The same is also true for matchings. An application is given to the enumeration of matchings with no $k$crossing (or with no $k$nesting).
 Publication:

arXiv Mathematics eprints
 Pub Date:
 January 2005
 arXiv:
 arXiv:math/0501230
 Bibcode:
 2005math......1230C
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Category Theory;
 05A18;
 05A15;
 05E10
 EPrint:
 Revision: page 8, revised Remark (3) page 11, revised Proof of Theorem 1. Page 20, revised Remark. page 23, added references