Enumeration of Stack-Sorting Preimages via a Decomposition Lemma

We give three applications of a recently-proven "Decomposition Lemma," which allows one to count preimages certain sets permutations under West's stack-sorting map $s$. first enumerate the permutation class $s^{-1}(\text{Av}(231,321))=\text{Av}(2341,3241,45231)$, finding new example an unbalanced Wilf equivalence. This result is equivalent enumeration sortable by ${\bf B}\circ s$, where B}$ bubble sort map. then prove that $s^{-1}(\text{Av}(231,312))$, $s^{-1}(\text{Av}(132,231))=\text{Av}(2341,1342,\underline{32}41,\underline{31}42)$, and $s^{-1}(\text{Av}(132,312))=\text{Av}(1342,3142,3412,34\underline{21})$ are counted so-called "Boolean-Catalan numbers," settling conjecture current author another Hossain. completes enumerations all form $s^{-1}(\text{Av}(\tau^{(1)},\ldots,\tau^{(r)}))$ for $\{\tau^{(1)},\ldots,\tau^{(r)}\}\subseteq S_3$ with exception set $\{321\}$. also find explicit formula $|s^{-1}(\text{Av}_{n,k}(231,312,321))|$, $\text{Av}_{n,k}(231,312,321)$ in $\text{Av}_n(231,312,321)$ $k$ descents. us conjectured identity involving Catalan numbers order ideals Young's lattice.