Bibliographic record
Abstract
We study permutations of the set [n] = {1, 2, . . . , n} written in cycle notation, for which each cycle forms an increasing or decreasing interval of positive integers. More generally, permutations whose cycle elements form arithmetic progressions are considered. We also investigate the class of generalised interval permutations, where each cycle can be rearranged in increasing order to form an interval of consecutive positive integers.. 1 Interval Permutations The partitions of the set [n] = {1, 2, . . . , n} into k nonempty subsets of consecutive integers are enumerated by ( n−1 k−1 ) since this is the number of ways of inserting k − 1 separators between the sequence of numbers 1, 2, . . . , n, 1 ≤ k ≤ n. We will obtain analogous results for permutations of [n], written in the cycle notation. Taking different orderings of the elements of the cycles into account gives several different analogues of the set partition case. Firstly, a permutation p of [n] will be called an interval permutation if every cycle of p consists of one increasing or decreasing sequence of consecutive integers. Even though the standard notation places the least member of a cycle in the first position, we adopt the convention to reckon only with the permutations in which the members of a cycle have been shifted so as to exhibit the maximal number of pairs of consecutive integers. An interval permutation is the unique member of its cycle class in which every v-cycle consists of v increasing or decreasing consecutive integers. Later in Section 5 we will also study the class of generalised interval permutations, where each cycle can be rearranged in increasing order to form an interval of consecutive integers. Denote the set of interval permutations of [n] with k cycles (also kpermutations below) by R(n, k), and let r(n, k) = |R(n, k)|. Also let r(n) = r(n, 1) + r(n, 2) + · · ·+ r(n, n).
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.024 | 0.006 |
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 source (direct Gemma or distilled Codex), 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".