Out of the parking lot and into the forest: Parking functions, bond lattices, and unimodal forests
Bibliographic record
Abstract
Rota introduced the bond lattice of a graph in [11] . It's a sublattice of the set partition lattice. For certain graphs, such as triangulation graphs, it's a sublattice of the important and oft studied noncrossing partition lattice. Parking functions are another central object in algebraic combinatorics. Stanley made the connection between them by defining a bijection from maximal chains of the noncrossing partition lattice to parking functions [14] . Motivated by Stanley's bijection, we study the maximal chains in the bond lattices of triangulation graphs. The number of maximal chains in the bond lattice of a triangulation graph is the number of ordered cycle decompositions [1] , as well as being the number of rooted unimodal forests [2] . In this paper, we find a recursive bijection between these maximal chains and rooted unimodal forests, based on a simpler recursion than that given in [1] .
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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 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 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; 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".