Parking functions, tree depth and factorizations of the full cycle into transpositions
Bibliographic record
Abstract
Consider the set of minimal factorizations of the canonical full cycle in the symmetric group on n + 1 symbols. In 2002, Biane found a remarkably simple bijection from this set to the set of parking functions of length n; the bijection maps a factorization to the sequence consisting of the smallest element from each transposition. Thus, it is utterly trivial to find the image of a factorization in this map, but reversing this map requires much more work. Furthermore, as far as parking functions are concerned, it appears that the largest element in each transposition can be discarded. We show, however, that the sequence given by the largest element of each transposition also displays some interesting properties. In particular, the natural area statistics on this sequence and the parking function together correspond to two natural statistics on trees: the inversion number (this is well known) and the non-inversion number. This allows us to present a bivariate generating series, which is a natural generalization of the univariate inversion series. We also give a number of related results.
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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 | 0.000 |
| 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".