Equivalence Relations of Permutations Generated by Constrained Transpositions
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Bibliographic record
Abstract
We consider a large family of equivalence relations on permutations in $S_n$ that generalise those discovered by Knuth in his study of the Robinson-Schensted correspondence. In our most general setting, two permutations are equivalent if one can be obtained from the other by a sequence of pattern-replacing moves of prescribed form; however, we limit our focus to patterns where two elements are transposed, conditional upon the presence of a third element of suitable value and location. For some relations of this type, we compute the number of equivalence classes, determine how many $n$-permutations are equivalent to the identity permutation, or characterise this equivalence class. Although our results include familiar integer sequences (e.g., Catalan, Fibonacci, and Tribonacci numbers) and special classes of permutations (layered, connected, and $123$-avoiding), some of the sequences that arise appear to be new. Nous considérons une famille de relations d’équivalence sur l'ensemble $S_n$ des permutations, qui généralisent les relations de Knuth liées à la correspondance Robinson-Schensted. Dans notre contexte général, deux permutations sont considérées comme équivalentes si l'une peut être obtenue de l'autre auprès d'une séquence de remplacements d'un motif par un autre selon des règles précisées. Désormais, nous ne considérons dans l’œuvre actuelle que les motifs qui correspondent à la transposition de deux éléments, conditionné sur la présence d'un élément de valeur et de position approprié. Pour plusieurs exemples de ce problème, nous énumérons les classes d'équivalence, nous déterminons combien de permutations sur $n$ éléments sont équivalentes à l'identité, ou nous précisons la forme des éléments dans cette dernière classe. Bien que nos résultats retrouvent des séquences des entiers très bien connues (nombres de Catalan, de Fibonacci, de Tribonacci...) ainsi que des classes de permutations déjà étudiées (en couches, connexes, sans motif $123$), nous trouvons également des séquences qui paraissent être nouvelles.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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