Structural equivalence and novelty in constrained tiling sequences: A computational classification framework

This research explores the relationship between combinatorial tiling problems and integer sequences, demonstrating how linear constraints on tile quantity and color lead to diverse and structurally rich sequences. While traditional sequences like the Fibonacci sequence and Narayana’s cows sequence arise from unrestricted tilings of strips with squares and k-minos, this paper introduces several restricted tiling models, including configurations where the number of squares equals the number of trominos (corresponding to OEIS A002426), the number of dominos equals the number of trominos (A387599), and color-balanced tilings where quantities of different tile types are constrained (e.g., A001850 and A098482), as well as variants with offset constraints such as requiring one more tile of a given type than another. To evaluate the structural novelty of these sequences, we pivot to a core thesis: if a sequence has multiple independent combinatorial realizations, it is typically not fundamentally new, but a member of a known structural class. We formalize a computational, partially heuristic classification procedure to determine when sequences are genuinely new versus when they are reinterpretations, and evaluate its performance over a 30-sequence benchmark dataset using standard classification metrics. Furthermore, we propose a heuristic asymptotic interpretation: in constrained tiling models, each independent linear constraint induces a Gaussian suppression factor, leading to asymptotic growth of the form O( λ n / n k / 2 ), consistent with known results in lattice-walk asymptotics and analytic combinatorics. Finally, we extend our framework to analyze the effects of modular constraints and local forbidden patterns, utilizing transfer-matrix methods to evaluate their distinct asymptotic behavior.