Pattern Avoidance in Reverse Double Lists

TL;DR

This paper studies pattern avoidance in reverse double lists, providing enumeration results for patterns of length up to 4, classifying Wilf classes, and analyzing growth for longer patterns.

Contribution

It introduces the concept of reverse double lists and fully classifies pattern avoidance for patterns up to length 4, also analyzing growth for longer patterns.

Findings

Enumerated reverse double lists avoiding patterns of length ≤ 4

Determined Wilf classes for these patterns

Characterized polynomial growth for patterns of length ≥ 5

Abstract

In this paper, we consider pattern avoidance in a subset of words on $\{1,1,2,2,\dots,n,n\}$ called reverse double lists. In particular a reverse double list is a word formed by concatenating a permutation with its reversal. We enumerate reverse double lists avoiding any permutation pattern of length at most 4 and completely determine the corresponding Wilf classes. For permutation patterns $\rho$ of length 5 or more, we characterize when the number of $\rho$-avoiding reverse double lists on $n$ letters has polynomial growth. We also determine the number of $1\cdots k$-avoiders of maximum length for any positive integer $k$.