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 $$\sum _{n\ge 0}\frac{t^n}{n!}\sum _{\sigma \in {\mathcal{N}\mathcal{M}}_n(\Gamma )}{x}^{\mathrm{LRmin}(\sigma )}{y}^{1+\mathrm{des}(\sigma )}$$ where $\Gamma$ is a set of permutations which start with 1 and have at most one descent, ${\mathcal{N}\mathcal{M}}_n(\Gamma )$ 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 $U_{\Gamma }(t,y)={\sum }_{n\ge 0}{U}_{\Gamma ,n}(y)\frac{t^n}{n!}$ 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 $p\ge 5$, or $\Gamma$ is the set of permutations $\sigma ={\sigma }_1\cdots {\sigma }_n$ of length $n=k_1+k_2$ where $k_1,k_2\ge 2$, ${\sigma }_1=1$, ${\sigma }_{k_1+1}=2$, and $\mathrm{des}(\sigma )=1$.

References in corpus (1)

• Anick-type resolutions and consecutive pattern avoidance