Generating trees growing on the left for pattern-avoiding inversion sequences

TL;DR

This paper introduces a new generating tree method for pattern-avoiding inversion sequences, providing algebraic generating functions for specific patterns and exploring broader applications.

Contribution

It presents a novel left-growing generating tree construction for inversion sequences, enabling exact computation of their algebraic generating functions for certain pattern avoidances.

Findings

Generated algebraic functions for sequences avoiding 201, 210, 010, and 102.

Provided a new combinatorial approach to pattern-avoiding inversion sequences.

Extended the method to more general pattern-avoidance settings.

Abstract

This work concerns a construction of pattern-avoiding inversion sequences from right to left we call the generating tree growing on the left. We first apply this construction to inversion sequences avoiding 201 and 210, resulting in a new way of computing their generating function. We then use a slightly modified construction to compute the generating function of inversion sequences avoiding 010 and 102, which was only conjectured before. These generating functions are algebraic in both instances. We end by discussing how the generating tree growing on the left can be applied in a more general setting.