Expected Patterns in Permutation Classes

TL;DR

This paper investigates the distribution of length-3 patterns within permutations avoiding the pattern 123, revealing exact counts and similarities to the 132-avoiding class, highlighting unexpected pattern regularities.

Contribution

It provides exact formulas for pattern occurrences in the class avoiding 123 and compares these with the 132-avoiding class, uncovering new pattern equivalences.

Findings

Number of 231 patterns is the same in both classes.

Exact formulas for pattern counts in Av(123).

Similarities between Av(123) and Av(132) classes.

Abstract

In the set of all patterns in $S_n$, it is clear that each k-pattern occurs equally often. If we instead restrict to the class of permutations avoiding a specific pattern, the situation quickly becomes more interesting. Mikl\'os B\'ona recently proved that, surprisingly, if we consider the class of permutations avoiding the pattern 132, all other non-monotone patterns of length 3 are equally common. In this paper we examine the class $\Av (123)$, and give exact formula for the occurrences of each length 3 pattern. While this class does not break down as nicely as $\Av (132)$, we find some interesting similarities between the two and prove that the number of 231 patterns is the same in each.