Enumeration of snakes and cycle-alternating permutations

Springer numbers are analogs of Euler numbers for the group of signed permutations. Arnol’d showed that they count some objects called snakes, which generalize alternating permutations. Hoffman established a link between Springer numbers, snakes, and some polynomials related with the successive derivatives of trigonometric functions. The goal of this article is to give further combinatorial properties of derivative polynomials, in terms of snakes and other objects: cyclealternating permutations, weighted Dyck or Motzkin paths, increasing trees and forests. We obtain some exponential generating functions in terms of trigonometric functions, and some ordinary generating functions in terms of J-fractions. We also define natural q-analogs, make a link with normal ordering problems and the combinatorial theory of differential equations.