$q$-analogs of rational numbers: from Ostrowski numeration systems to perfect matchings
Bibliographic record
Abstract
We consider the $q$-deformation of rational numbers introduced recently by Morier-Genoud and Ovsienko. We propose three enumerative interpretations of these $q$-rationals: in terms of a new version of Ostrowski's numeration system for integers, in terms of order ideals of fence posets and in terms of perfect matchings of snake graphs. Contrary to previous results which are restricted to rational numbers greater than one, our interpretations work for all positive rational numbers and are based on a single combinatorial object for defining both the numerator and denominator. The proofs rest on order-preserving bijections between posets over these objects. We recover a formula for a $q$-analog of Markoff numbers. We also deduce a fourth interpretation given in terms of the integer points inside a polytope in $\mathbb{R}^k$ on both sides of a hyperplane where $k$ is the length of the continued fraction expansion.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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; 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".