Bijections Between {\\L}ukasiewicz Walks and Generalized Tandem Walks
Bibliographic record
Abstract
In this article, we study the enumeration by length of several walk models on\nthe square lattice. We obtain bijections between walks in the upper half-plane\nreturning to the $x$-axis and walks in the quarter plane. A recent work by\nBostan, Chyzak, and Mahboubi has given a bijection for models using small\nnorth, west, and south-east steps. We adapt and generalize it to a bijection\nbetween half-plane walks using those three steps in two colours and a\nquarter-plane model over the symmetrized step set consisting of north,\nnorth-west, west, south, south-east, and east. We then generalize our\nbijections to certain models with large steps: for given $p\\geq1$, a bijection\nis given between the half-plane and quarter-plane models obtained by keeping\nthe small south-east step and replacing the two steps north and west of length\n1 by the $p+1$ steps of length $p$ in directions between north and west. This\nmodel is close to, but distinct from, the model of generalized tandem walks\nstudied by Bousquet-M\\'elou, Fusy, and Raschel.\n
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.002 | 0.002 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.001 | 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; both teacher heads agree on what is shown here.
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".