Enumerate remaining rectangulation families in Merino–Mütze (2023) Table 3

Determine the enumeration—i.e., the counting sequences and, when possible, the generating functions—for the families of pattern-avoiding rectangulations identified in Merino and Mütze (2023, Table 3) whose enumeration remains open, by computing the number of non-equivalent rectangulations of size n for each such family defined by a fixed finite set of forbidden rectangulation patterns.

Background

The paper resolves ten guillotine diagonal cases and additional non-guillotine families (vortex rectangulations and whirls), confirming several previously stated conjectures and deriving algebraic generating functions.

However, Merino and Mütze’s Table 3 lists further families of rectangulations defined by forbidding specific small patterns, and the authors explicitly note that the enumeration of some of these families remains open. They suggest that variants of their methods could handle some of these cases, with results to appear in an extended version.