From order one catalytic decompositions to context-free specifications: the rewiring bijection

TL;DR

This paper offers a combinatorial approach to derive algebraic solutions for order one catalytic equations, enabling the construction of context-free specifications and bijections for combinatorial structures.

Contribution

It provides a new combinatorial derivation for algebraicity of solutions to order one catalytic equations, facilitating the creation of simple tree-based specifications.

Findings

Algebraic solutions for order one catalytic equations are obtainable via combinatorial methods.

The approach yields context-free specifications for derivation trees of combinatorial structures.

Unified framework for classical combinatorial enumeration problems involving catalytic equations.

Abstract

A celebrated result of Bousquet-M\'elou and Jehanne states that the bivariate power series solutions of so-called combinatorial polynomial equations with one catalytic variable, also known as catalytic equations, are algebraic series. We give a purely combinatorial derivation of this result in the case of order one catalytic equations (those involving only one univariate unknown series). In particular our approach provides a tool to produce context-free specifications, or bijections with simple multi-type families of trees, for the derivation trees of combinatorial structures that are directly governed by an order one catalytic decomposition. This provides a simple unified framework to deal with various combinatorial interpretation problems that were solved or raised over the last 50 years since the first such catalytic equation was written by W. T. Tutte in the late 60's to enumerate rooted planar maps.