  We extend the reciprocity method of Jones and Remmel to study generating
functions of the form $$\sum_{n \geq 0} \frac{t^n}{n!} \sum_{\sigma \in
\mathcal{NM}_n(\Gamma)}x^{\mathrm{LRmin}(\sigma)}y^{1+\mathrm{des}(\sigma)}$$
where $\Gamma$ is a set of permutations which start with 1 and have at most one
descent, $\mathcal{NM}_n(\Gamma)$ is the set of permutations $\sigma$ in the
symmetric group $\mathfrak{S}_n$ which have no $\Gamma$-matches,
$\mathrm{des}(\sigma)$ is the number of descents of $\sigma$ and
$\mathrm{LRmin}(\sigma)$ 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 $U_{\Gamma}(t,y) = \sum_{n\geq
0}U_{\Gamma,n}(y) \frac{t^n}{n!}$ and the coefficients $U_{\Gamma,n}(y)$
satisfy some simple recursions in the case where $\Gamma$ equals
$\{1324,123\}$, $\{1324 \cdots p,12 \cdots (p-1)\}$ for $p \geq 5$, or $\Gamma$
is the set of permutations $\sigma = \sigma_1 \cdots \sigma_n$ of length
$n=k_1+k_2$ where $k_1,k_2 \geq 2$, $\sigma_1 =1$, $\sigma_{k_1+1}=2$, and
$\mathrm{des}(\sigma) =1$.
