Completing the enumeration of inversion sequences avoiding one or two patterns of length 3

TL;DR

This paper completes the enumeration of inversion sequences avoiding one or two patterns of length 3 by introducing new constructions, including generating trees and shifted inversion sequences, and analyzes their asymptotic behavior.

Contribution

It provides the first complete enumeration of such inversion sequences and introduces shifted inversion sequences as a new tool for combinatorial enumeration.

Findings

Enumeration of 24 classes of pattern-avoiding inversion sequences completed

Introduction of shifted inversion sequences for broader combinatorial analysis

Discussion of exponential and super-exponential asymptotic behaviors

Abstract

We present four constructions of inversion sequences, and use them to compute the enumeration sequences of 24 classes of pattern-avoiding inversion sequences. This completes the enumeration of inversion sequences avoiding one or two patterns of length 3. Some of our constructions are based on generating trees. Others involve pattern-avoiding words, which we also count using generating trees. To solve some of these cases, we introduce a generalization of inversion sequences, which we call shifted inversion sequences. Lastly, we briefly discuss the asymptotics of pattern-avoiding inversion sequences, focusing on their exponential or super-exponential behavior.