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

Abstract

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 $\mathrm{des}(\sigma) =1$.