Asymptotics of Pattern Avoidance in the Permutation-Tuple and Klazar Set Partition Settings

TL;DR

This paper investigates the asymptotic behavior of pattern avoidance in set partitions and permutation tuples, extending previous results and proposing new conjectures, with a complete solution for the growth rates of certain avoidance classes.

Contribution

It extends the classification of set partitions avoiding patterns and solves the asymptotics of parallel permutation pattern avoidance within exponential factors.

Findings

Classified set partitions with subexponential growth in avoidance counts

Solved asymptotics of parallel permutation pattern avoidance

Proposed several new conjectures in pattern avoidance theory

Abstract

We consider asymptotics of set partition pattern avoidance in the sense of Klazar. One of the results of this paper extends work of Alweiss, and finds a classification for set partitions $\pi$ such that the number of set partitions of $[n]$ avoiding $\pi$ grows more slowly than $n^{cn}$ for all $c>0$. Several conjectures are proposed, and the related question of asymptotics of parallel ($k$-tuple) permutation pattern avoidance is considered and solved completely to within an exponential factor, generalizing Marcus and Tardos's 2004 proof of the Stanley-Wilf Conjecture.