Pattern Avoidance in Poset Permutations

TL;DR

This paper generalizes pattern avoidance from permutations on totally ordered sets to those on partially ordered sets, establishing inequalities and classifications for such permutations.

Contribution

It extends the concept of pattern avoidance to poset permutations and classifies posets where specific pattern avoidance counts are equal.

Findings

Proves that $Av_P(132) \,\leq\, Av_P(123)$ for any poset $P$.

Classifies posets where $Av_P(132) = Av_P(123)$ holds.

Abstract

We extend the concept of pattern avoidance in permutations on a totally ordered set to pattern avoidance in permutations on partially ordered sets. The number of permutations on $P$ that avoid the pattern $\pi$ is denoted $Av_P(\pi)$. We extend a proof of Simion and Schmidt to show that $Av_P(132) \leq Av_P(123)$ for any poset $P$, and we exactly classify the posets for which equality holds.