Consecutive patterns in inversion sequences II: avoiding patterns of relations

TL;DR

This paper explores the avoidance of consecutive patterns of relations in inversion sequences, revealing connections to well-known sequences like Catalan and Fibonacci numbers, and classifies these patterns into Wilf equivalence classes.

Contribution

It introduces the study of consecutive relation patterns in inversion sequences, classifies them into Wilf equivalence classes, and proves several conjectures and enumeration formulas.

Findings

Enumeration of inversion sequences avoiding certain relation patterns yields Catalan, Fibonacci, and polynomial numbers.

Classification of relation patterns into Wilf equivalence classes based on avoidance counts.

Provides bijective proofs and confirms conjectures related to pattern avoidance and enumeration.

Abstract

Inversion sequences are integer sequences $e=e_{1}e_{2}\dots e_{n}$ such that $0\leq e_{i}<i$ for each $i$. The study of patterns in inversion sequences was initiated by Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck in the classical (non-consecutive) case, and later by Auli--Elizalde in the consecutive case, where the entries of a pattern are required to occur in adjacent positions. In this paper we continue this investigation by considering {\em consecutive patterns of relations}, in analogy to the work of Martinez--Savage in the classical case. Specifically, given two binary relations $R_{1},R_2\in\{\leq,\geq,<,>,=,\neq\}$, we study inversion sequences $e$ with no subindex $i$ such that $e_{i}R_{1}e_{i+1}R_{2}e_{i+2}$. By enumerating such inversion sequences according to their length, we obtain well-known quantities such as Catalan numbers, Fibonacci numbers and central polynomial numbers, relating inversion sequences to other combinatorial structures. We also classify consecutive patterns of relations into Wilf equivalence classes, according to the number of inversion sequences avoiding them, and into more restrictive classes that consider the positions of the occurrences of the patterns. As a byproduct of our techniques, we obtain a simple bijective proof of a result of Baxter--Shattuck and Kasraoui about Wilf-equivalence of vincular patterns, and we prove a conjecture of Martinez and Savage, as well as related enumeration formulas for inversion sequences satisfying certain unimodality conditions.