Restricted 132-Dumont permutations

A permutation $\pi$ is said to be {\em Dumont permutations of the first kind} if each even integer in $\pi$ must be followed by a smaller integer, and each odd integer is either followed by a larger integer or is the last element of $\pi$ (see, for example, \cite{Z}). In \cite{D} 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 exactly once) an arbitrary pattern on $k$ letters. In several interesting cases the generating function depends only on $k$.Comment: 12 page

Restricted Dumont permutations, Dyck paths, and noncrossing partitions

We complete the enumeration of Dumont permutations of the second kind avoiding a pattern of length 4 which is itself a Dumont permutation of the second kind. We also consider some combinatorial statistics on Dumont permutations avoiding certain patterns of length 3 and 4 and give a natural bijection between 3142-avoiding Dumont permutations of the second kind and noncrossing partitions that uses cycle decomposition, as well as bijections between 132-, 231- and 321-avoiding Dumont permutations and Dyck paths. Finally, we enumerate Dumont permutations of the first kind simultaneously avoiding certain pairs of 4-letter patterns and another pattern of arbitrary length.Comment: 20 pages, 5 figure

Enumeration of Dumont permutations avoiding certain four-letter patterns

In this paper, we enumerate Dumont permutations of the fourth kind avoiding or containing certain permutations of length 4. We also conjecture a Wilf-equivalence of two 4-letter patterns on Dumont permutations of the first kind