Completing the enumeration of inversion sequences avoiding triples of relations

TL;DR

This paper completes the enumeration of inversion sequences avoiding specific triples of binary relations, providing generating functions and asymptotic analysis for these classes, advancing the understanding of pattern avoidance in combinatorics.

Contribution

It fully enumerates all 14 previously uncounted classes of pattern-avoiding inversion sequences defined by triples of relations, using novel generating tree methods.

Findings

Derived algebraic generating functions for many classes.

Established asymptotic behavior of the counting sequences.

Provided a complete enumeration for all 14 classes.

Abstract

An inversion sequence of length $n$ is an integer sequence $(a_1, \ldots, a_n)$ such that $0 \le a_i < i$ for all $i$. The study of pattern-avoiding inversion sequences was initiated in 2015 by Mansour and Shattuck and in 2016 by Corteel, Martinez, Savage and Weselcouch. Martinez and Savage later defined a new type of pattern, a triple of binary relations, of which there are currently 14 uncounted avoidance classes. We complete the enumeration for all of these classes using generating tree methods "growing on the left" and "growing on the right". For many of these classes we are able to find algebraic generating functions. We also discuss the asymptotic behaviour of the counting sequences.