[Paper Review] Enumeration of snakes and cycle-alternating permutations

Matthieu Josuat-Vergès|arXiv (Cornell University)|Nov 3, 2010

Advanced Combinatorial Mathematics20 references20 citations

TL;DR

This paper provides combinatorial interpretations of derivative polynomials associated with trigonometric functions, linking them to signed permutations (snakes), cycle-alternating permutations, weighted Dyck and Motzkin paths, and increasing trees/forests. It establishes generating functions in terms of trigonometric series and J-fractions, introduces q-analogs, and connects the results to normal ordering problems and differential equations, offering a unified combinatorial framework for Springer numbers and related sequences.

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.

Motivation & Objective

• To extend the combinatorial understanding of Springer numbers S_n beyond their known generating function by linking them to multiple combinatorial objects.
• To provide bijective and algebraic proofs for generating functions of derivative polynomials P_n(t), Q_n^{(a)}(t), and R_n(t) using combinatorial structures.
• To introduce and study q-analogs of these polynomials and connect them to normal ordering problems in quantum mechanics and differential equations.
• To unify various combinatorial families—snakes, cycle-alternating permutations, weighted lattice paths, and increasing trees—under a common generating function framework.

Proposed method

• Derives recurrence relations and generating functions for derivative polynomials P_n(t), Q_n^{(a)}(t), and R_n(t) using differential identities and trigonometric expansions.
• Establishes bijections between increasing trees and snakes via reading words and sign assignments based on leaf positions.
• Constructs weighted Dyck and Motzkin paths that encode the coefficients of Q_n^{(a)}(t), with weights derived from path structure and descent patterns.
• Uses J-fraction continued fractions to express ordinary generating functions, proving them combinatorially via path decompositions.
• Introduces q-analogs of the polynomials and relates them to normal ordering of operators in the context of differential equations.
• Applies the combinatorial theory of differential equations to interpret the polynomials as counting weighted trees and forests.

Experimental results

Research questions

• RQ1 How can the derivative polynomials P_n(t) and Q_n^{(a)}(t) be interpreted combinatorially in terms of signed permutations and lattice paths?
• RQ2 What is the combinatorial meaning of the parameter t in the context of snakes and cycle-alternating permutations?
• RQ3 Can the generating functions of Q_n^{(a)}(t) be interpreted via continued fractions (J-fractions) in a bijective way, and how does this relate to known analytic results?
• RQ4 How do q-analogs of the Springer numbers and derivative polynomials arise from normal ordering problems in differential equations?
• RQ5 What is the structural relationship between increasing trees, forests, and the various permutation classes like snakes and cycle-alternating permutations?

Key findings

• The exponential generating function for Q_n^{(a)}(t) is given by (cos z - t sin z)^{-a}, generalizing the known result for a=1.
• The ordinary generating function for Q_n^{(a)}(t) is expressed as a J-fraction (continued fraction), providing a combinatorial interpretation of a classically known analytic identity.
• A bijection is constructed between increasing trees and snakes, where the sign assignment depends on the number of empty leaves before each node in the reading word.
• The polynomial R_n(t) = Q_n^{(2)}(t) satisfies R_n(1) = 2^n E_{n+1}, linking it to Euler numbers and β-snakes.
• The q-analog of the derivative polynomials is defined and connected to normal ordering problems, extending known q-analogs of Euler numbers.
• Cycle-alternating permutations are shown to be the image of snakes under Foata’s fundamental transform, providing a new combinatorial model for the same sequence.