Enumeration of snakes and cycle-alternating permutations

Springer numbers are an analog of Euler numbers for the group of signed permutations. Arnol'd showed that they count some objects called snakes, that 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 the generating functions, in terms of trigonometric functions for exponential ones and in terms of J-fractions for ordinary ones. We also define natural q-analogs, make a link with normal ordering problems and combinatorial theory of differential equations.

[1]  Sergi Elizalde,et al.  Cycle-up-down permutations , 2011, Australas. J Comb..

[2]  D. Foata,et al.  Multivariable Tangent and Secant q-derivative Polynomials , 2013, 1304.2486.

[3]  I. Goulden,et al.  Combinatorial Enumeration , 2004 .

[4]  Sylvie Corteel,et al.  The Matrix Ansatz, orthogonal polynomials, and permutations , 2010, Adv. Appl. Math..

[5]  R. Stanley.  A Survey of Alternating Permutations , 2009, 0912.4240.

[6]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[7]  Pierre Leroux,et al.  Combinatorial resolution of systems of differential equations. IV. separation of variables , 1988, Discret. Math..

[8]  Rene F. Swarttouw,et al.  Hypergeometric Orthogonal Polynomials , 2010 .

[9]  Michael E. Hoffman.  Derivative Polynomials, Euler Polynomials, and Associated Integer Sequences , 1999, Electron. J. Comb..

[10]  Philippe Flajolet,et al.  COMBINATORIAL MODELS OF CREATION{ANNIHILATION , 2010, 1010.0354.

[11]  M. Lothaire.  Combinatorics on words: Bibliography , 1997 .

[12]  Matthieu Josuat-Vergès,et al.  Combinatorics of the Three-Parameter PASEP Partition Function , 2009, Electron. J. Comb..

[13]  Sylvie Corteel.  Crossings and alignments of permutations , 2007, Adv. Appl. Math..

[14]  William Y. C. Chen,et al.  Labeled Ballot Paths and the Springer Numbers , 2010, SIAM J. Discret. Math..

[15]  D. Dumont.  Further Triangles of Seidel-Arnold Type and Continued Fractions Related to Euler and Springer Numbers , 1995 .

[16]  Gérard Viennot,et al.  Combinatorial resolution of systems of differential equations, I. Ordinary differential equations , 1986 .

[17]  Michael E. Hoffman.  DERIVATIVE POLYNOMIALS FOR TANGENT AND SECANT , 1995 .

[18]  Sylvie Corteel,et al.  Crossings of signed permutations and q-Eulerian numbers of type B , 2013 .

[19]  Vladimir I. Arnold,et al.  The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups , 1992 .

[20]  Donald E. Knuth,et al.  Computation of Tangent, Euler, and Bernoulli Numbers* , 1967 .

[21]  Philippe Flajolet.  Combinatorial aspects of continued fractions , 1980, Discret. Math..

[22]  Shi-Mei Ma,et al.  Derivative polynomials and enumeration of permutations by number of interior and left peaks , 2012, Discret. Math..

[23]  M. Purtill.  André permutations, lexicographic shellability and the cd-index of a convex polytope , 1993 .

[24]  D. Foata.  CHAPTER 16 – Nombres d'Euler et Permutations Alternantes , 1973 .

[25]  Jiang Zeng,et al.  Un autre q-analogue des nombres d’Euler , 2001 .