Permutations Almost Avoiding Monotone Distant Patterns

TL;DR

This paper explores permutation classes avoiding certain monotone distant patterns, establishing Wilf-Equivalences and providing new bounds on their exponential growth rates, thus advancing understanding of pattern avoidance with gap constraints.

Contribution

It introduces new Wilf-Equivalences among permutation classes and derives bounds on growth rates for monotone distant pattern avoidance.

Findings

Identified large classes of Wilf-Equivalences between permutation classes.

Established new bounds on exponential growth rates for monotone distant patterns.

Connected previous pattern avoidance results to distant pattern frameworks.

Abstract

In a previous work, B\'ona and Pantone studied permutations that avoided all but one pattern of length $k$ that began with a length $k-1$ increasing subsequence. We draw the connection between that idea and distant patterns, first discussed heavily in a work by Dimitrov, and study similar permutation classes, where the index not part of the increasing subsequence can vary. We find a large class of Wilf-Equivalences between $k+1$ classes of $k$ patterns of length $k+1$, and outline several classes of unbalanced Wilf-Equivalences related to the first class. Using this, we are also find new bounds on the exponential growth rate on all monotone distant patterns with a single gap constraint.