Thron-type continued fractions (T-fractions) for some classes of increasing trees

TL;DR

This paper introduces new classes of increasing labeled trees that provide combinatorial interpretations for certain Thron-type continued fractions with quasi-affine coefficients, extending previous affine cases.

Contribution

It extends the combinatorial interpretation of Thron-type continued fractions to quasi-affine coefficients using novel tree classes and bijections, building on classical bijections and recent work.

Findings

New classes of increasing trees with combinatorial interpretations

Bijections between trees and labeled paths like Motzkin and Schröder

Conjecture on vincular pattern distribution in permutations

Abstract

We introduce some classes of increasing labeled and multilabeled trees, and we show that these trees provide combinatorial interpretations for certain Thron-type continued fractions with coefficients that are quasi-affine of period 2. Our proofs are based on bijections from trees to labeled Motzkin or Schr\"oder paths; these bijections extend the well-known bijection of Fran\c{c}on--Viennot (1979) interpreted in terms of increasing binary trees. This work can also be viewed as a sequel to the recent work of Elvey Price and Sokal (2020), where they provide combinatorial interpretations for Thron-type continued fractions with coefficients that are affine. Towards the end of the paper, we conjecture an equidistribution of vincular patterns on permutations.