Generating functions for descents over permutations which avoid sets of consecutive patterns

We extend the reciprocity method of Jones and Remmel to study generating functions of the form $$\sum _{n\ge 0}\frac{t^n}{n!}\sum _{\sigma \in {\mathcal{N}\mathcal{M}}_n(\mathrm{\Gamma })}{x}^{\mathrm{L}\mathrm{R}\mathrm{m}\mathrm{i}\mathrm{n}(\sigma )}{y}^{1+\mathrm{d}\mathrm{e}\mathrm{s}(\sigma )}$$ where $\mathrm{\Gamma }$ is a set of permutations which start with 1 and have at most one descent, ${\mathcal{N}\mathcal{M}}_n(\mathrm{\Gamma })$ is the set of permutations $\sigma$ in the symmetric group ${\mathfrak{S}}_n$ which have no $\mathrm{\Gamma }$-matches, $\mathrm{d}\mathrm{e}\mathrm{s}(\sigma )$ is the number of descents of $\sigma$ and $\mathrm{L}\mathrm{R}\mathrm{m}\mathrm{i}\mathrm{n}(\sigma )$ is the number of left-to-right minima of $\sigma$. We show that this generating function is of the form $\left( \frac{1}{U_{\Gamma}(t,y)}\right)^x$where$U_{\Gamma}(t,y) = \sum_{n\geq 0}U_{\Gamma,n}(y) \frac{t^n}{n!}$andthecoefficients$U_{\Gamma,n}(y)$satisfysomesimplerecursionsinthecasewhere$\Gamma$equals${1324,123}$,${1324 \cdots p,12 \cdots (p-1)}$for$p \geq 5$,or$\Gamma$isthesetofpermutations$\sigma = \sigma_1 \cdots \sigma_n$oflength$n=k_1+k_2$where$k_1,k_2 \geq 2$,$\sigma_1 =1$,$\sigma_{k_1+1}=2$,and$$\mathrm{d}\mathrm{e}\mathrm{s}$(\sigma) =1$.