On the enumeration of permutations avoiding chains of patterns

In 2019, Bóna and Smith introduced the notion of strong pattern avoidance, saying that a permutation $\pi$ strongly avoids a pattern $\sigma$ if $\pi$ and ${\pi }^2$ both avoid $\sigma$. Recently, Archer and Geary generalized the idea of strong pattern avoidance to chain avoidance, in which a permutation $\pi$ avoids a chain of patterns $({\tau }^{(1)}:{\tau }^{(2)}:\cdots :{\tau }^{(k)})$ if the $i$-th power of the permutation avoids the pattern ${\tau }^{(i)}$ for $1\le i\le k$. In this paper, we give explicit formulae for the number of sets of permutations avoiding certain chains of patterns. Our results give affirmative answers to two conjectures proposed by Archer and Geary.