Parity of transversals of Latin squares
Bibliographic record
Abstract
We introduce a notion of parity for transversals, and use it to show that in Latin squares of order 2 mod 4 , the number of transversals is a multiple of 4. We also demonstrate a number of relationships (mostly congruences modulo 4) involving E 1 , ⋯ , E n , where E i is the number of diagonals of a given Latin square that contain exactly i different symbols. Let A ( i ∣ j ) denote the matrix obtained by deleting row i and column j from a parent matrix A . Define t i j to be the number of transversals in L ( i ∣ j ) , for some fixed Latin square L . We show that t a b ≡ t c d mod 2 for all a , b , c , d and L . Also, if L has odd order then the number of transversals of L equals t a b mod 2. We conjecture that t a c + t b c + t a d + t b d ≡ 0 mod 4 for all a , b , c , d . In the course of our investigations we prove several results that could be of interest in other contexts. For example, we show that the number of perfect matchings in a k -regular bipartite graph on 2 n vertices is divisible by 4 when
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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 | 0.012 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.001 | 0.005 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.016 | 0.002 |
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".