Rectangulations avoiding a pattern
Kaoru Sano
TL;DR
This work analyzes strong rectangulations avoiding a fixed geometric pattern and proves a uniform exponential deficit in growth rate relative to all strong rectangulations. By translating rectangulations into leftmost history quadrant walks and employing a pattern-insertion technique, the authors derive a generating-function bound that yields a growth-constant gap: for any pattern P of size L, $\limsup \#\mathrm{Rec}(n;P)^{1/n} \le \Lambda - 1/\Lambda^{3L-1}$ with $\Lambda = 27/2$. Consequently, the proportion of $P$-avoiding rectangulations vanishes as $n$ grows. The results establish the first uniform exponential upper bound for pattern-avoiding rectangulations and outline avenues for tightening the bound and determining explicit growth constants across sizes.
Abstract
Fix a strong rectangulation pattern $P$ of size $L$. We show that the growth constant of the class of strong rectangulations avoiding $P$ is strictly smaller than $Λ=27/2$, the growth constant for all strong rectangulations. More precisely, forbidding any such $P$ yields a pattern-uniform exponential drop of at least $Λ- 1/Λ^{3L-1}$. Consequently, the proportion of $P$-avoiding rectangulations among all rectangulations tends to zero as $n\to \infty$. This is the first result on the uniform drop of exponential growth for pattern-avoiding rectangulations. The proof utilizes the standard correspondence with leftmost history quadrant walks, along with a pattern-insertion scheme that controls the radius of convergence of the associated generating functions, thereby establishing the first uniform exponential upper bound for rectangulation classes defined by geometric avoidance.