论文标题
集群和$ r $ stirling排列的统计数据
Statistics on clusters and $r$-Stirling permutations
论文作者
论文摘要
Goulden $ \ unicode {x2013} $ Jackson群集方法，适用于Elizalde和Noy的排列，通过出现规定的连续模式来计算排列的问题，以计算群集的计数，这些模式与很多结构相关。最近，Zhuang发现了群集方法的概括，该方法专门通过其他排列统计数据进行了修改，即逆下降数$ \ permatatorName {ides} $，逆峰号$ \ operatatorName {ipk} $，以及近左峰值$ \ prepatateRemNMUMEN $ \ perperAtatorName $ \ perperatoRearname {iilpk {iilpk} $。继续进行这项工作，我们研究了$ 2134 \ cdots m $ -clusters by $ \ operatatorName {ides} $，$ \ perpotatorname {ipk} $，$ \ operatatorName {ipk} $，以及$ \ operatotorname {ilpk} $，这允许我们通过计数$ 21的$ 213的unterys $ 统计数据。模式$ 12 \ cdots（M-2）M（M-1）$的类似结果是通过对称参数获得的。在此过程中，我们发现$ 2134 \ cdots（r+1）$ - 集群与Gessel和Stanley介绍的$ R $ stirling排列相等，我们在这两个排列家庭之间建立了一些联合等分。
The Goulden$\unicode{x2013}$Jackson cluster method, adapted to permutations by Elizalde and Noy, reduces the problem of counting permutations by occurrences of a prescribed consecutive pattern to that of counting clusters, which are special permutations with a lot of structure. Recently, Zhuang found a generalization of the cluster method which specializes to refinements by additional permutation statistics, namely the inverse descent number $\operatorname{ides}$, the inverse peak number $\operatorname{ipk}$, and the inverse left peak number $\operatorname{ilpk}$. Continuing this line of work, we study the enumeration of $2134\cdots m$-clusters by $\operatorname{ides}$, $\operatorname{ipk}$, and $\operatorname{ilpk}$, which allows us to derive formulas for counting permutations by occurrences of the consecutive pattern $2134\cdots m$ jointly with each of these statistics. Analogous results for the pattern $12\cdots (m-2)m(m-1)$ are obtained via symmetry arguments. Along the way, we discover that $2134\cdots (r+1)$-clusters are equinumerous with $r$-Stirling permutations introduced by Gessel and Stanley, and we establish some joint equidistributions between these two families of permutations.
