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 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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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; a candidate call from one teacher head, 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".