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Bibliographic record
Abstract
We introduce the primitive Eulerian polynomial \(P_{\mathcal{A}}(z)\) of a central hyperplane arrangement \(\mathcal{A}\). It is a reparametrization of its cocharacteristic polynomial. Previous work by the first author implicitly shows that for simplicial arrangements, \(P_{\mathcal{A}}(z)\) has nonnegative coefficients. For reflection arrangements of types A and B, the same work interprets the coefficients of \(P_{\mathcal{A}}(z)\) using the (flag) excedance statistic on (signed) permutations.The main result of this article is to provide an interpretation of the coefficients of \(P_{\mathcal{A}}(z)\) for all simplicial arrangements using only the geometry and combinatorics of \(\mathcal{A}\). This new interpretation sheds more light to the case of reflection arrangements and, for the first time, gives combinatorial significance to the coefficients of the primitive Eulerian polynomial of the reflection arrangement of type D, for which no well-behaved excedance statistic is known. In type B, we establish a link between the primitive Eulerian polynomial and the \(1/2\)-Eulerian polynomial of Savage and Viswanathan (2012). We present some results and conjectures regarding the real-rootedness of \(P_{\mathcal{A}}(z)\).Mathematics Subject Classifications: 52C35, 05A05Keywords: Hyperplane arrangement, Eulerian polynomial, Tits product, permutation statistics
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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 | 0.000 | 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 it