AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 60(3) (2014), Pages 279–305 Enumeration of snakes and cycle-alternating permutations Matthieu Josuat-Verges` ∗ Fakult¨at f¨ur Mathematik Universit¨at Wien, 1090 Wien Austria
[email protected] To the memory of Vladimir Arnol’d Abstract 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: cycle- alternating 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. 1 Introduction It is well-known that the Euler numbers En defined by ∞ zn En =tanz +secz (1) n! n=0 count alternating permutations in Sn, i.e. σ such that σ1 >σ2 <σ3 > ...σn. The study of these, as well as other classes of permutations counted by En,isavast topic in enumerative combinatorics, see the survey of Stanley [27]. By a construction due to Springer [26], there is an integer K(W ) defined for any finite Coxeter group ∗ Supported by the French National Research Agency ANR, grant ANR08-JCJC-0011, and the Austrian Science Foundation FWF, START grant Y463 M.