Bibliographic record
Abstract
The sequence a1 · · · am is a common subsequence in the set of permutations S = {π1,..., πk} on [n] if it is a subsequence of πi(1) · · · πi(n) and πj(1) · · · πj(n) for some distinct πi, πj ∈ S. Recently, Beame and Huynh-Ngoc (2008) showed that when k ≥ 3, every set of k permutations on [n] has a common subsequence of length at least n 1/3. We show that, surprisingly, this lower bound is asymptotically optimal for all constant values of k. Specifically, we show that for any k ≥ 3 and n ≥ k 2 there exists a set of k permutations on [n] in which the longest common subsequence has length at most 32(kn) 1/3. The proof of the upper bound is constructive, and uses elementary algebraic techniques. ∗ Research supported by NSF grants CCF-0514870 and CCF-0830626 † Research supported by a scholarship from the Fonds québécois de la recherche sur la nature et les technologies (FQRNT). ‡ Research supported by NSF grant CCF-0830626 and a Vietnam Education Foundation Fellowship 1 1
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How this classification was reachedexpand
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 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
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 teacher head, 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".