Let $S_n$ denote the group all permutations of $n$. For every permutation $σ$, we let $\mathrm{des}(σ)$ denote the number of descents in $σ$ and $\mathrm{LRMin}(σ)$ denote the number of left-to-right minima of $σ$. Given a sequence $τ= τ_1 \cdots τ_n$ of distinct positive integers, we define the reduction of $τ$, $\mathrm{red}(τ)$, to be the permutation of $S_n$ that results by replacing the $i$-th smallest element of $τ$ by $i$. If $Γ$ is a set of permutations, we say that a permutation $σ= σ_1 \ldots σ_n \in S_n$ has a $Γ$-match starting at position $i$ if there is a $i < j$ such that $\mathrm{red}(σ_i σ_{i+1} \ldots σ_j) \in Γ$. We let $Γ$-$\mathrm{mch}(σ)$ denote the number of $Γ$-matches in $σ$. We let $\mathcal{NM}_n(Γ)$ be the set of $σ\in S_n$ such that $Γ$-$\mathrm{mch}(σ) = 0$. In this paper, we modify Jones and Remmel's reciprocity method to study the generating function of the form \begin{equation} \mbox{NM}_Γ(t,x,y)=\sum_{n \geq 0} \frac{t^n}{n!} \mbox{NM}_{Γ,n}(x,y) \end{equation} where $\displaystyle \mbox{NM}_{Γ,n}(x,y) =\sum_{σ\in \mathcal{NM}_n(Γ)}x^{\mathrm{LRmin}(σ)}y^{1+\mathrm{des}(σ)}$ in the case where we no longer insist that all the permutations $τ\in Γ$ have at most one descent.