Bibliographic record
Abstract
Lattice paths are fundamental combinatorial classes and they appear in many guises in physics, queuing theory, language theory, and pure mathematics. New formulas for the asymptotic enumeration of walks restricted to a quadrant have appeared in the past ten years, mostly driven by novel systematic, and analytic approaches. In this tutorial we will present the universe of lattice path classes, and survey some of the strategies used to deduce enumerative formulas. The generating functions for several models of walks in cones satisfy nice differential equations, and algorithms to find, manipulate, and solve differential equations have been central to this study. We will discuss how to express the generating functions as diagonals of multivariate rational functions, and how to reconcile the results of different approaches to this computation.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.004 | 0.004 |
| Science and technology studies | 0.002 | 0.007 |
| Scholarly communication | 0.008 | 0.015 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.002 | 0.006 |
| Insufficient payload (model declined to judge) | 0.011 | 0.002 |
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 source (direct Gemma or distilled Codex), 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".