Enumeration of snakes and cycle-alternating permutations

TL;DR

This paper explores the combinatorial structures related to Springer numbers, including snakes and cycle-alternating permutations, and derives generating functions and q-analogs connecting these objects with derivatives of trigonometric functions.

Contribution

It introduces new combinatorial properties of derivative polynomials linked to Springer numbers, expanding understanding of their structure and generating functions.

Findings

Derived generating functions in terms of trigonometric functions

Established connections with normal ordering problems

Defined natural q-analogs and related combinatorial objects

Abstract

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.