Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials

Abstract

We find a Thron-type continued fraction (T-fraction) for the ordinary generating function of the Ward polynomials, as well as for some generalizations employing a large (indeed infinite) family of independent indeterminates. Our proof is based on a bijection between super-augmented perfect matchings and labeled Schröder paths, which generalizes Flajolet's bijection between perfect matchings and labeled Dyck paths.

arXiv commentLaTeX2e, 36 pages (includes 4 figures). Version 2 corrects a small error in the definition of crossing number (p. 6) and includes a proof of the previously conjectured (1.25)/(1.26)

References in corpus (4)

• Bivariate Generating Functions for a Class of Linear Recurrences: General Structure
• Generalized Stirling permutations and forests: Higher-order Eulerian and Ward numbers
• De transformatione seriei divergentis 1 - mx + m(m+n)x^2 - m(m+n)(m+2n)x^3 + etc. in fractionem continuam
• Comparing two statistical ensembles of quadrangulations: a continued fraction approach

Cited by in corpus (1)

• A simple algorithm for expanding a power series as a continued fraction