  We show that the bistatistic of right nestings and right crossings in
matchings without left nestings is equidistributed with the number of
occurrences of two certain patterns in permutations, and furthermore that this
equidistribution holds when refined to positions of these statistics in
matchings and permutations. For this distribution we obtain a non-commutative
generating function which specializes to Zagier's generating function for the
Fishburn numbers after abelianization.
  As a special case we obtain proofs of two conjectures of Claesson and
Linusson.
  Finally, we conjecture that our results can be generalized to involving left
crossings of matchings too.
