Permutations avoiding sets of patterns with long monotone subsequences

TL;DR

This paper studies the enumeration of permutations that avoid specific pattern sets with long monotone subsequences, providing insights into their size relative to classical monotone pattern avoidance classes.

Contribution

It introduces a new enumeration approach for permutations avoiding all but one of certain pattern sets with long monotone subsequences, expanding understanding of permutation class sizes.

Findings

Enumeration formulas for these permutation classes

Comparison with classical monotone pattern avoidance classes

Insights into the structure of permutations with long monotone subsequences

Abstract

We enumerate permutations that avoid all but one of the $k$ patterns of length $k$ starting with a monotone increasing subsequence of length $k-1$. We compare the size of such permutation classes to the size of the class of permutations avoiding the monotone increasing subsequence of length $k-1$.