On ascent sequences avoiding 021 and a pattern of length four

TL;DR

This paper extends the enumeration of ascent sequences avoiding the pattern 021 to those avoiding 021 and a length-four pattern, classifying Wilf-equivalence, deriving generating functions, and providing new combinatorial interpretations.

Contribution

It introduces a comprehensive enumeration of ascent sequences avoiding 021 and any length-four pattern, identifying Wilf-equivalence classes and deriving explicit generating functions.

Findings

Enumeration formulas for ascent sequences avoiding 021 and length-four patterns

Classification of Wilf-equivalence classes for these avoidance sets

New combinatorial interpretations for OEIS entries

Abstract

Ascent sequences of length $n$ avoiding the pattern $021$ are enumerated by the $n$-th Catalan number $C_n=\frac{1}{n+1}\binom{2n}{n}$. In this paper, we extend this result and enumerate ascent sequences avoiding $\{021,\tau\}$, where $\tau$ is a pattern of length four. We in turn identify all of the corresponding Wilf-equivalence classes and find generating function formulas corresponding to each class. In a couple of cases, we make use of an auxiliary statistic and the kernel method to ascertain the generating function. In several cases, our work of enumeration is shortened by establishing the equivalence of $\{021,\tau\}$- and $\{021,\tau'\}$-avoiders of a given length through an explicit bijection. As a consequence of our results, one obtains new combinatorial interpretations in terms of ascent sequences for several of the entries in the OEIS.