Counting Runs of Ones with Overlapping Parts in Binary Strings Ordered Linearly and Circularly
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
On a binary $(0-1)$ string of length $n$ the $\ell$-overlapping counting scheme of runs of $1$ s of a fixed length $k$ is considered. According to this scheme, a run of 1s of length $k$ which is counted may have overlapping part of length at most $\ell$, $0\leq \ell<k\leq n$, with the previous run of $1$ s of length $k$ that has been enumerated. The numbers of all $\ell$-overlapping runs of $1$ s of length $k$ in all $2^{n}$ binary strings (linearly or circularly ordered) of length $n$ are examined, and simple and easy to compute closed explicit expressions are provided via the probability mass function and the expected value of properly defined random variables. The numbers of binary strings of length $n$, ordered on a line or on a circle, with a specific number of $\ell$-overlapping runs of 1s of length $k$ are also provided via closed expressions. The numbers which are studied, are potentially useful in several scientific areas like applied probability, engineering and bioinformatics. The study is illustrated by extensive numerical examples.
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 | 0.000 | 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 | 0.001 |
| 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