Combinatorics and Asymptotics of Positive Systems of Linear Catalytic Equations

TL;DR

This paper analyzes positive linear systems with one catalytic variable, revealing their combinatorial structure and universal asymptotic behavior, with applications in lattice paths and permutation counting.

Contribution

It provides a complete combinatorial and asymptotic analysis of these systems, linking generating functions to context-free grammars and establishing universal asymptotics.

Findings

Generating functions satisfy positive polynomial systems linked to context-free grammars.

Established universal asymptotic behavior for these systems.

Applications include lattice path and permutation enumeration.

Abstract

We provide a complete combinatorial and asymptotic analysis of positive linear systems of equations in one catalytic variable that appear in several combinatorial problems such as in lattice path counting or stack-sortable permutation counting. We show that the corresponding generating functions satisfy a positive polynomial system of equations (which is associated to a context-free grammar). Furthermore we prove a universal asymptotic behaviour.