A NEW NOTION OF TRANSITIVITY FOR GROUPS AND SETS OF PERMUTATIONS
Bibliographic record
Abstract
Let Ω = {1, 2, …, n} where n ⩾ 2. The shape of an ordered set partition P = (P1, …, Pk) of Ω is the integer partition λ = (λ1, …, λk) defined by λi = |Pi|. Let G be a group of permutations acting on Ω. For a fixed partition λ of n, we say that G is λ-transitive if G has only one orbit when acting on partitions P of shape λ. A corresponding definition can also be given when G is just a set. For example, if λ = (n − t, 1, …, 1), then a λ-transitive group is the same as a t-transitive permutation group, and if λ = (n − t, t), then we recover the t-homogeneous permutation groups. We use the character theory of the symmetric group Sn to establish some structural results regarding λ-transitive groups and sets. In particular, we are able to generalize a celebrated result of Livingstone and Wagner [Math. Z. 90 (1965) 393–403] about t-homogeneous groups. We survey the relevant examples coming from groups. While it is known that a finite group of permutations can be at most 5-transitive unless it contains the alternating group, we show that it is possible to construct a nontrivial t-transitive set of permutations for each positive integer t. We also show how these ideas lead to a combinatorial basis for the Bose–Mesner algebra of the association scheme of the symmetric group and a design system attached to this association scheme.
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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 | 0.002 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.004 | 0.003 |
| Science and technology studies | 0.003 | 0.008 |
| Scholarly communication | 0.005 | 0.012 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.007 | 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 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".