Upper bounds for the Stanley-Wilf limit of 1324 and other layered patterns

TL;DR

This paper establishes upper bounds on the Stanley-Wilf limits for layered permutation patterns and the specific pattern 1324, linking these bounds to pattern length and proposing conjectures about permutation classes.

Contribution

It provides new upper bounds for the Stanley-Wilf limits of layered patterns and the pattern 1324, and introduces conjectures relating permutation avoidance and inversion counts.

Findings

Stanley-Wilf limit for layered patterns is at most 4ℓ².

Stanley-Wilf limit for pattern 1324 is at most 16.

If conjectures hold, the limit for 1324 is at most approximately 13.002.

Abstract

We prove that the Stanley-Wilf limit of any layered permutation pattern of length $\ell$ is at most $4\ell^2$, and that the Stanley-Wilf limit of the pattern 1324 is at most 16. These bounds follow from a more general result showing that a permutation avoiding a pattern of a special form is a merge of two permutations, each of which avoids a smaller pattern. If the conjecture is true that the maximum Stanley-Wilf limit for patterns of length $\ell$ is attained by a layered pattern then this implies an upper bound of $4\ell^2$ for the Stanley-Wilf limit of any pattern of length $\ell$. We also conjecture that, for any $k\ge 0$, the set of 1324-avoiding permutations with $k$ inversions contains at least as many permutations of length $n+1$ as those of length $n$. We show that if this is true then the Stanley-Wilf limit for 1324 is at most $e^{\pi\sqrt{2/3}} \simeq 13.001954$.