The problem of Permutation Pattern Matching (PPM) asks, given a permutation $π:[n]\to[n]$, viewed as the point set $P=\{(i,π(i)):i\in[n]\}$, and a pattern $σ$ of small length $k$, whether $P$ contains an order-isomorphic copy of $σ$. The PPM problem and its (exact and approximate) counting variants have been studied extensively in the literature. We consider a preprocessing-query version: after preprocessing $P$, one must determine whether a query $P'\subseteq P$ contains $σ$, where $P'$ is obtained by intersecting $P$ with a geometric shape (e.g., an axis-aligned rectangle or $k$ such rectangles). This naturally generalizes the classical range query problem in computational geometry, which corresponds to the case $|σ|=1$. We extend the study of this problem from patterns of length $2$ to patterns of length $3$ (and in some cases beyond), and prove new fine-grained equivalences, separations, and applications: (1) Single rectangle: For all patterns of length $3$, detection can be done with $\tilde{O}(n)$ preprocessing and $\tilde{O}(1)$ query time. The proof goes through a new tool of independent interest: reductions from rectangle queries to a simpler family of slab queries. (2) Multiple rectangles: For $σ=132$, detection is equivalent to the online set disjointness problem; we also show connections to online $k$-set disjointness. To our knowledge, this is the first equivalence between a PPM-type problem and online set disjointness, a problem playing a fundamental role in fine-grained complexity. (3) Counting: As applications, we obtain an $O(n^{1.725})$ time algorithm for exact counting of $5$-patterns based on the rectangle-to-slab reduction, improving upon [Beniamini and Lavee, ICALP 2025], and an $\tilde{O}(n^{7/4})$ approximate counting algorithm for $6$-patterns, the first to break the $O(n^2)$ barrier for $6$-patterns.

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