  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.
