Bidirectional Algorithms for Polygon Triangulations and (m + 2)-Angulations via Fuss–Catalan Numbers
Bibliographic record
Abstract
Polygon triangulations and their generalizations to m+2−angulations are fundamental in combinatorics and computational geometry. This paper presents a unified linear-time framework that establishes explicit bijections between m−Dyck words, planted m+1−ary trees, and m+2−angulations of convex polygons. We introduce stack-based and tree-based algorithms that enable reversible conversion between symbolic and geometric representations, prove their correctness and optimal complexity, and demonstrate their scalability through extensive experiments. The approach reveals a hierarchical decomposition encoded by Fuss–Catalan numbers, providing a compact and uniform representation for triangulations, quadrangulations, pentangulations, and higher-arity angulations. Experimental comparisons show clear advantages over rotation-based methods in both runtime and memory usage. The framework offers a general combinatorial foundation that supports efficient enumeration, compressed representation, and extensions to higher-dimensional or non-convex settings.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".