Subregularity in infinitely labeled generating trees of restricted permutations

TL;DR

This paper introduces a new method leveraging subregularities in generating trees to solve permutation pattern avoidance problems, especially where existing algorithms like FinLabel fail to find finite solutions.

Contribution

It proposes a unified approach using subregularities in generating trees to derive systems of equations for permutation classes, extending beyond FinLabel's limitations.

Findings

The new procedure successfully derives systems of equations for complex permutation classes.

It demonstrates the effectiveness of subregularities in generating trees for enumeration.

The approach can be implemented computationally to solve previously intractable problems.

Abstract

In this paper, we revisit the application of generating trees to the pattern avoidance problem for permutations. In particular, we study this problem for certain general sets of patterns and propose a new procedure leveraging the FinLabel algorithm and exploiting the subregularities in the associated generating trees. We consider some general kinds of generating trees for which the FinLabel algorithm fails to determine in a finite number of iterations the generating function that enumerates the underlying class of permutations. Our procedure provides a unified approach in these cases leading to a system of equations satisfied by a certain finite set of generating functions which can be readily solved with the aid of programming.