Wilf classification of subsets of four letter patterns

Let $\mathcal{S}_n$ be the symmetric group of all permutations of $\{1,\ldots,n\}$, and let $w_k$ be the number of distinct Wilf classes of subsets of exactly $k$ patterns in $\mathcal{S}_4$. We show that $w_{10}=10624$, $w_{11}=5857$, $w_{12}=3044$, $w_{13}=1546$, $w_{14}=786$, $w_{15}=393$, $w_{16}=198$, $w_{17}=105$, $w_{18}=55$, $w_{19}=28$, $w_{20}=14$, $w_{21}=8$, $w_{22}=4$, $w_{23}=2$, and $w_{24}=1$.