Bibliographic record
Abstract
We study the generation of the fixed ideals of finite partial orders with particular attention in finding Gray codes for their generation. Pruesse and Ruskey conjecture that the graph J(P,k), which contains as vertices the k-ideals of the poset P, with an edge between vertices that differ by a swap, has a Hamiltonian path. The conjecture is true for series-parallel posets and interval orders. We prove the conjecture also holds for the fence posets, but that the conjecture is false for the 3-ideals of the crown poset with six elements. We also provide an infinite family of posets for which the conjecture does not hold. We study the Whitney numbers of fence posets to show a different but related conjecture of Pruesse and Ruskey also holds for fences and crowns with a small number of exceptions. We study Hamiltonian cycles in the Johnson graph (n,k) with certain properties about adjacent elements which we name t-full Hamiltonian cycles. We show their connection to the recently proved Middle Levels Theorem and how to construct them. We use t-full Hamiltonian cycles to show that, for k <= n-2 the graph J(Cr(2n), k) has a Hamiltonian cycle. We also apply 2-full Hamiltonian cycles to show that (P,3) has a Hamiltonian path for any height two poset P. Further, we introduce t-full Hamiltonian connected paths, and show that 1-full Hamiltonian connected paths exist in J(n,k). We introduce a weaker version of t-full Hamiltonian paths called pair-adjacent paths, and give algorithms for their construction. We show how these algorithms can generate the 3-ideals for crown posets with more than 6 elements. Finally, we study two applications of the generation of fixed-sized ideals. We formulate the information set decoding method in terms of error-correcting codes with a poset metric, and show how the probability of success in the guessing phase of such algorithms is minimized for anti-chain posets. We generalize ordered covering arrays (OCA) for general posets, and give elementary constructions of OCAs for level-regular rooted tree posets.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 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 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".