Some open problems on permutation patterns

TL;DR

This survey discusses open problems in permutation pattern research, focusing on enumeration, asymptotics, algebraic properties, and growth rates of permutation classes, highlighting recent developments and unresolved questions.

Contribution

It provides a focused overview of unresolved issues in permutation patterns not covered in recent literature, emphasizing the pattern 1324 and related algebraic and combinatorial topics.

Findings

Recent progress on the enumeration of pattern 1324

Upper bounds for avoiders of patterns of length k

Insights into the algebraic and topological properties of permutation posets

Abstract

This is a brief survey of some open problems on permutation patterns, with an emphasis on subjects not covered in the recent book by Kitaev, \emph{Patterns in Permutations and words}. I first survey recent developments on the enumeration and asymptotics of the pattern 1324, the last pattern of length 4 whose asymptotic growth is unknown, and related issues such as upper bounds for the number of avoiders of any pattern of length $k$ for any given $k$. Other subjects treated are the M\"obius function, topological properties and other algebraic aspects of the poset of permutations, ordered by containment, and also the study of growth rates of permutation classes, which are containment closed subsets of this poset.