Generating functions for permutations which avoid consecutive patterns with multiple descents

Abstract

Let $S_n$ denote the group all permutations of $n$. For every permutation $\sigma$, we let $\mathrm{des}(\sigma)$ denote the number of descents in $\sigma$ and $\mathrm{LRMin}(\sigma)$ denote the number of left-to-right minima of $\sigma$. Given a sequence $\tau = \tau_1 \cdots \tau_n$ of distinct positive integers, we define the reduction of $\tau$, $\mathrm{red}(\tau)$, to be the permutation of $S_n$ that results by replacing the $i$-th smallest element of $\tau$ by $i$. If $\Gamma$ is a set of permutations, we say that a permutation $\sigma = \sigma_1 \ldots \sigma_n \in S_n$ has a $\Gamma$-match starting at position $i$ if there is a $i < j$ such that $\mathrm{red}(\sigma_i \sigma_{i+1} \ldots \sigma_j) \in \Gamma$. We let $\Gamma$-$\mathrm{mch}(\sigma)$ denote the number of $\Gamma$-matches in $\sigma$. We let $\mathcal{NM}_n(\Gamma)$ be the set of $\sigma \in S_n$ such that $\Gamma$-$\mathrm{mch}(\sigma) = 0$. In this paper, we modify Jones and Remmel's reciprocity method to study the generating function of the form \begin{equation} \mbox{NM}_{\Gamma}(t,x,y)=\sum_{n \geq 0} \frac{t^n}{n!} \mbox{NM}_{\Gamma,n}(x,y) \end{equation} where $\displaystyle \mbox{NM}_{\Gamma,n}(x,y) =\sum_{\sigma \in \mathcal{NM}_n(\Gamma)}x^{\mathrm{LRmin}(\sigma)}y^{1+\mathrm{des}(\sigma)}$ in the case where we no longer insist that all the permutations $\tau \in \Gamma$ have at most one descent.