A NEW NOTION OF TRANSITIVITY FOR GROUPS AND SETS OF PERMUTATIONS
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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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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| 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) | 0.000 | 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