Distinct flattened partitions avoiding a pattern of length four

Let $\mathcal{P}_n$ denote the set of distinct permutations of length $n$ that arise from the flattening process applied to the partitions of $[n]=\{1,\ldots,n\}$. In this paper, we consider the problem of avoidance of a single classical pattern of length four by members of $\mathcal{P}_n$. Let $p_n