Restricted 132-Dumont permutations

A permutation π is said to be a Dumont permutation of the first kind if each even integer in π must be followed by a smaller integer, and each odd integer is either followed by a larger integer or is the last element of π (see, for example, www.theory.csc.uvic.ca/∼cos/inf/perm/Genocchi Info.html). In Duke Math. J. 41 (1974), 305–318, Dumont showed that certain classes of permutations on n letters are counted by the Genocchi numbers. In particular, Dumont showed that the (n+ 1)st Genocchi number is the number of Dummont permutations of the first kind on 2n letters. In this paper we study the number of Dumont permutations of the first kind on n letters avoiding the pattern 132 and avoiding (or containing exac tly once) an arbitrary pattern on k letters. In several interesting cases the generating function depends only on k.