Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Given a permutation $\pi=\pi_1\pi_2\cdots \pi_n \in S_n$, we say an index $i$ is a peak if $\pi_{i-1} < \pi_i > \pi_{i+1}$. Let $P(\pi)$ denote the set of peaks of $\pi$. Given any set $S$ of positive integers, define ${P_S(n)=\{\pi\in S_n:P(\pi)=S\}}$. Billey-Burdzy-Sagan showed that for all fixed subsets of positive integers $S$ and sufficiently large $n$, $|P_S(n)|=p_S(n)2^{n-|S|-1}$ for some polynomial $p_S(x)$ depending on $S$. They conjectured that the coefficients of $p_S(x)$ expanded in a binomial coefficient basis centered at $max(S)$ are all positive. We show that this is a consequence of a stronger conjecture that bounds the modulus of the roots of $p_S(x)$. Furthermore, we give an efficient explicit formula for peak polynomials in the binomial basis centered at $0$, which we use to identify many integer roots of peak polynomials along with certain inequalities and identities. Etant donné une permutation $\pi=\pi_1\pi_2\cdots \pi_n \in S_n$ du groupe symétrique, nous disons qu’un indice i est unsommet si $\pi_{i-1} < \pi_i > \pi_{i+1}$. Soit $P(\pi)$ l’ensemble des sommets de $\pi$. Billey-Burdzy-Sagan ont montré que,pour tout sous-ensemble d’entiers positifs S et n suffisamment grand, le nombre de permutations de $n$ éléments avecensemble de sommets $S$ est $|P_S(n)|=p_S(n)2^{n-|S|-1}$ pour un certain polynôme $p_S(x)$ dépendant de $S$.. Ils ont fait la conjectureque les coefficients du polynôme $p_S(x)$ exprimé dans une base de coefficients binomiaux centrée en $max(S)$ sont touspositifs. Nous montrons que cela découle d’une conjecture plus forte qui borne le module des racines du polynôme$p_S(x)$. De plus, nous donnons une formule explicite efficace pour les polynômes sommets dans la base binomialecentrée en $0$, que nous utilisons pour identifier plusieurs racines entières de polynômes sommets, ainsi que certainesinégalités et identités.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.003 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.001 |
| 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