Stanley–Wilf-type exponential bound for pattern-avoiding rectangulations

Ascertain whether, for every family of rectangulations defined by a fixed finite set of forbidden rectangulation patterns, the number of non-equivalent rectangulations of size n is bounded above by A^n for some constant A independent of n, thereby establishing a Stanley–Wilf-like exponential bound for pattern-avoiding rectangulations.

Background

The authors propose investigating broader classes obtained by forbidding additional local patterns in rectangulations and ask whether such classes exhibit uniform exponential growth bounds, analogous to the Stanley–Wilf conjecture for permutations.

This question connects structural avoidance in rectangulations to asymptotic enumeration, seeking a universal exponential bound across all fixed-pattern-avoidance families.