Bibliographic record
Abstract
A Dyck path is a lattice path in the first quadrant using steps $u=(1,1)$ and $d=(1,-1)$, starting at the origin and ending on the $x$-axis. A Dyck sojourn is a Dyck path with only one return to the $x$-axis. Various subclasses of Dyck paths are shown to be counted by Catalan numbers: Dyck paths with weakly highest first sojourn and semilength $n+1$ are counted by twice the $n$th Catalan number and Dyck paths with semilength $n+1$ where the second sojourn is weakly highest are counted by the $n$th Catalan number. Also Dyck paths in which the first sojourn is strictly highest are equinumerous to Dyck paths with exactly one peak of maximum height. Some of these equivalences are proved using bijections. Generating functions for all these subclasses are also developed in the paper. The asymptotics as $n\to\infty$ for the case where the first sojourn is strictly highest are explored.
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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 | 0.002 | 0.012 |
| Meta-epidemiology (narrow) | 0.002 | 0.002 |
| Meta-epidemiology (broad) | 0.003 | 0.001 |
| Bibliometrics | 0.001 | 0.004 |
| Science and technology studies | 0.002 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.001 |
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".