Generating functions for permutations which avoid consecutive patterns with multiple descents

TL;DR

This paper develops a generating function approach for counting permutations avoiding certain consecutive patterns with multiple descents, extending previous methods to more general cases.

Contribution

It modifies Jones and Remmel's reciprocity method to handle permutations avoiding patterns with multiple descents, broadening the scope of pattern avoidance enumeration.

Findings

Derived new generating functions for permutations avoiding multiple-descent patterns.

Extended existing methods to accommodate patterns with multiple descents.

Provided formulas for counting permutations based on left-to-right minima and 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.