A bijection between the sets of $(a,b,b^2)$-generalized Motzkin paths avoiding $\mathrm{uvv}$-patterns and $\mathrm{uvu}$-patterns
Bibliographic record
Abstract
A generalized Motzkin path, called G-Motzkin path for short, of length $n$ is a lattice path from $(0, 0)$ to $(n, 0)$ in the first quadrant of the XY-plane that consists of up steps $\mathrm{u}=(1, 1)$, horizontal steps $\mathrm{h}=(1, 0)$, vertical steps $\mathrm{v}=(0, -1)$ and down steps $\mathrm{d}=(1, -1)$. An $(a,b,c)$-G-Motzkin path is a weighted G-Motzkin path such that the $\mathrm{u}$-steps, $\mathrm{h}$-steps, $\mathrm{v}$-steps and $\mathrm{d}$-steps are weighted respectively by $1, a, b$ and $c$.Let $\tau$ be a word on $\{\mathrm{u}, \mathrm{h}, \mathrm{v}, \mathrm{d}\}$, denote by $\mathcal{G}_n^{\tau}(a,b,c)$ the set of $\tau$-avoiding $(a,b,c)$-G-Motzkin paths of length $n$ for a pattern $\tau$. In this paper, we consider the $\mathrm{uvv}$-avoiding $(a,b,c)$-G-Motzkin paths and provide a direct bijection $\sigma$ between $\mathcal{G}_n^{\mathrm{uvv}}(a,b,b^2)$ and $\mathcal{G}_n^{\mathrm{uvu}}(a,b,b^2)$. Finally, the set of fixed points of $\sigma$ is also described and counted.
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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 | 0.004 | 0.011 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.002 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| 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; 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".