Asymptotic results on Klazar set partition avoidance

TL;DR

This paper derives asymptotic bounds for the number of set partitions avoiding a specific pattern, providing precise exponential estimates for block cases and exploring graph-theoretic reformulations with conjectures.

Contribution

It introduces asymptotic bounds for partition avoidance, including a general lower bound and a graph-theoretic perspective, advancing understanding in combinatorial pattern avoidance.

Findings

Established asymptotic bounds for pattern-avoiding partitions

Provided exponential accuracy for block case

Proposed conjectures in graph-theoretic reformulation

Abstract

We establish asymptotic bounds for the number of partitions of $[n]$ avoiding a given partition in Klazar's sense, obtaining the correct answer to within an exponential for the block case. This technique also enables us to establish a general lower bound. Additionally, we consider a graph theoretic restatement of partition avoidance problems, and propose several conjectures.