Pattern Avoidance in Ordered Set Partitions

TL;DR

This paper studies the enumeration of ordered set partitions that avoid specific permutation patterns, providing exact counts for certain cases and asymptotic growth rates, advancing combinatorial understanding of pattern avoidance.

Contribution

It offers new enumeration formulas for avoiding patterns in ordered set partitions and establishes asymptotic growth rate results, including a Stanley-Wilf type theorem.

Findings

Exact enumeration for avoiding permutation 12

Enumeration of 3-block and n-1 block avoiding partitions

Asymptotic growth rates with Stanley-Wilf type results

Abstract

In this paper we consider the enumeration of ordered set partitions avoiding a permutation pattern of length 2 or 3. We provide an exact enumeration for avoiding the permutation 12. We also give exact enumeration for ordered partitions with 3 blocks and ordered partitions with n-1 blocks avoiding a permutation of length 3. We use enumeration schemes to recursively enumerate 123-avoiding ordered partitions with any block sizes. Finally, we give some asymptotic results for the growth rates of the number of ordered set partitions avoiding a single pattern; including a Stanley-Wilf type that exhibits existence of such growth rates.