Non-crossing linked partitions, the partial order << on NC(n), and the S-transform
Why this work is in the frame
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Bibliographic record
Abstract
The paper establishes a connection between two recent combinatorial developments in free probability: the non-crossing linked-partitions introduced by Dykema in 2007 to study the S-transform, and the partial order�on NC n introduced by Belinschi and Nica in 2008 in order to study relations between free and Boolean probability. More precisely, one has a canonical bijection between NCL n (the set of all noncrossing linked-partitions of 1,...,n) and the set α, β�α, β NC n, α�β. As a consequence of this bijection, one gets an alternative description of Dykema’s formula expressing the moments of a noncommutative random variable a in terms of the coefficients of the reciprocal S-transform 1 Sa. Moreover, due to the Boolean features of�, this formula can be simplified to a form which resembles the moment-cumulant formula from c-free probability. 1.
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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 | 0.001 | 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 | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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 it