Pattern avoidance in revised ascent sequences

TL;DR

This paper introduces revised ascent sequences, a new class of integer sequences related to Cayley permutations, and explores their pattern avoidance properties using combinatorial bijections and generating functions.

Contribution

It defines revised ascent sequences, establishes a bijection with ascent sequences, and analyzes pattern avoidance to derive enumerative results.

Findings

Bijection between ascent and revised ascent sequences

Enumeration of pattern-avoiding revised ascent sequences

Development of generating functions and kernel method applications

Abstract

Inspired by the definition of modified ascent sequences, we introduce a new class of integer sequences called revised ascent sequences. These sequences are defined as Cayley permutations where each entry is a leftmost occurrence if and only if it serves as an ascent bottom. We construct a bijection between ascent sequences and revised ascent sequences by adapting the classic hat map, which transforms ascent sequences into modified ascent sequences. Additionally, we investigate revised ascent sequences that avoid a single pattern, leading to a wealth of enumerative results. Our main techniques include the use of bijections, generating trees, generating functions, and the kernel method.