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 $$\sum_{n \geq 0} \frac{t^n}{n!} \sum_{σ\in \mathcal{NM}_n(Γ)}x^{\mathrm{LRmin}(σ)}y^{1+\mathrm{des}(σ)}$$ where $Γ$ is a set of permutations which start with 1 and have at most one descent, $\mathcal{NM}_n(Γ)$ is the set of permutations $σ$ in the symmetric group $\mathfrak{S}_n$ which have no $Γ$-matches, $\mathrm{des}(σ)$ is the number of descents of $σ$ and $\mathrm{LRmin}(σ)$ is the number of left-to-right minima of $σ$. We show that this generating function is of the form $\left( \frac{1}{U_Γ(t,y)}\right)^x$ where $U_Γ(t,y) = \sum_{n\geq 0}U_{Γ,n}(y) \frac{t^n}{n!}$ and the coefficients $U_{Γ,n}(y)$ satisfy some simple recursions in the case where $Γ$ equals $\{1324,123\}$, $\{1324 \cdots p,12 \cdots (p-1)\}$ for $p \geq 5$, or $Γ$ is the set of permutations $σ= σ_1 \cdots σ_n$ of length $n=k_1+k_2$ where $k_1,k_2 \geq 2$, $σ_1 =1$, $σ_{k_1+1}=2$, and $\mathrm{des}(σ) =1$.