Statistics on clusters and $r$-Stirling permutations

Abstract: 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.