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 $\sigma ={\sigma }_1\dots {\sigma }_n\in S_n$ has a $\Gamma$-match starting at position $i$ if there is a $i<j$ such that $\mathrm{red}({\sigma }_i{\sigma }_{i+1}\dots {\sigma }_j)\in \Gamma$. We let $\Gamma$-$\mathrm{mch}(\sigma )$ denote the number of $\Gamma$-matches in $\sigma$. We let ${\mathcal{N}\mathcal{M}}_n(\Gamma )$ 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}_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 $\tau \in \Gamma$ have at most one descent.

arXiv admin note: substantial text overlap with arXiv:1510.07190

References in corpus (1)

• Anick-type resolutions and consecutive pattern avoidance