Permutations that strongly avoid 132

TL;DR

This paper studies permutations that avoid the pattern 132 in both the permutation and its square, providing enumeration results and confirming that their growth rate is 2.

Contribution

It introduces the concept of strong pattern avoidance and proves the growth rate for permutations strongly avoiding 132 is 2.

Findings

Enumerates permutations of size n strongly avoiding 132

Proves the growth rate of such permutations is 2

Confirms a conjecture about their asymptotic behavior

Abstract

A permutation $\pi$ strongly avoids the pattern $\tau$ if both $\pi$ and $\pi^2$ avoid $\tau$. In this paper, we enumerate permutations of size $n$ that strongly avoid the pattern 132. This enumeration allows us to prove a conjecture that the growth rate of such permutations is 2.