Inversion sequences avoiding pairs of patterns

TL;DR

This paper systematically studies inversion sequences avoiding pairs of length-3 patterns, revealing connections to classical combinatorial objects and solving a conjecture about specific pattern avoidance.

Contribution

It provides a comprehensive enumeration of inversion sequences avoiding two patterns of length 3, extending previous work and solving an open conjecture.

Findings

Connections to OEIS sequences and classical combinatorial objects

Enumeration of inversion sequences avoiding specific pattern pairs

Resolution of a conjecture on (021,120)-avoiding sequences

Abstract

The enumeration of inversion sequences avoiding a single pattern was initiated by Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck independently. Their work has sparked various investigations of generalized patterns in inversion sequences, including patterns of relation triples by Martinez and Savage, consecutive patterns by Auli and Elizalde, and vincular patterns by Lin and Yan. In this paper, we carried out the systematic study of inversion sequences avoiding two patterns of length $3$. Our enumerative results establish further connections to the OEIS sequences and some classical combinatorial objects, such as restricted permutations, weighted ordered trees and set partitions. Since patterns of relation triples are some special multiple patterns of length $3$, our results complement the work by Martinez and Savage. In particular, one of their conjectures regarding the enumeration of $(021,120)$-avoiding inversion sequences is solved.