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

TL;DR

This paper investigates the asymptotic behavior of pattern avoidance in set partitions and permutation-tuples, providing growth rate classifications and extending existing notions of pattern avoidance.

Contribution

It derives asymptotics for the number of pattern-avoiding set partitions and permutation-tuples, generalizing previous concepts and establishing growth classifications.

Findings

Asymptotics of pattern avoidance are characterized within an exponential factor.

A classification of growth rates for set partition pattern classes is provided.

A new notion of permutation-tuple avoidance is introduced and analyzed.

Abstract

We consider asymptotics of set partition pattern avoidance in the sense of Klazar. Our main result derives the asymptotics of the number of set partitions avoiding a given set partition within an exponential factor, which leads to a classification of possible growth rates of set partition pattern classes. We further define a notion of permutation-tuple avoidance, which generalizes notions of Aldred et al. and the usual permutation pattern setting, and similarly determine the number of permutation-tuples avoiding a given tuple to within an exponential factor.