Some polynomials associated with up-down permutations

Up-down permutations, introduced many years ago by André under the name alternating permutations, were studied by Carlitz and coauthors in a series of papers in the 1970s. We return to this class of permutations and discuss several sets of polynomials associated with them. These polynomials allow us to divide up-down permutations into various subclasses, with the aid of the exponential formula. We find explicit, albeit complicated, expressions for the coefficients, and we explain how one set of polynomials counts up-down permutations of even length when evaluated at x=1, and of odd length when evaluated at x=2. We also introduce a new kind of sequence that is equinumerous with the up-down permutations, and we give a bijection.