  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).
