Generating functions for descents over permutations which avoid sets of consecutive patterns

TL;DR

This paper develops a method to compute generating functions for permutations avoiding certain consecutive patterns, focusing on descents and left-to-right minima, with explicit formulas for specific pattern sets.

Contribution

It extends the reciprocity method to derive explicit generating functions for permutations avoiding complex pattern sets with descent and minima statistics.

Findings

Generated explicit formulas for specific pattern sets

Derived recursive relations for coefficients in the generating functions

Connected pattern avoidance with descent and minima statistics

Abstract

We extend the reciprocity method of Jones and Remmel to study generating functions of the form $$\sum_{n \geq 0} \frac{t^n}{n!} \sum_{\sigma \in \mathcal{NM}_n(\Gamma)}x^{\mathrm{LRmin}(\sigma)}y^{1+\mathrm{des}(\sigma)}$$ where $\Gamma$ is a set of permutations which start with 1 and have at most one descent, $\mathcal{NM}_n(\Gamma)$ 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 $U_{\Gamma}(t,y) = \sum_{n\geq 0}U_{\Gamma,n}(y) \frac{t^n}{n!}$ 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 $p \geq 5$, or $\Gamma$ is the set of permutations $\sigma = \sigma_1 \cdots \sigma_n$ of length $n=k_1+k_2$ where $k_1,k_2 \geq 2$, $\sigma_1 =1$, $\sigma_{k_1+1}=2$, and $\mathrm{des}(\sigma) =1$.