A matrix approach to the enumeration of naturally labeled posets
Bibliographic record
Abstract
We propose a matrix approach for generating naturally labeled posets by representing each poset $P$ on the set $[n]$ as a Boolean poset matrix $A$. This algebraic representation enables a systematic handling of partial orderings through matrix extensions $A^v$. We show that $A^v$ defines a valid poset matrix if and only if the Boolean vector $v\in\mathbb B^n$ represents an order ideal of the poset $P$ associated to $A$, equivalently satisfying the fixed-point equation $vA=v$. Based on this characterization, we develop a sieve algorithm that generates all admissible extension vectors efficiently. Furthermore, we explore the twin-class decomposition of $A$, which partitions the elements of $P$ according to identical down- and up-sets. This structure provides an algebraic foundation for Burnside-type enumeration for Birkhoff's question on counting nonisomorphic posets on $[n]$ through the automorphism group ${\rm Aut}(A)$. Finally, we present an algorithmic generation scheme for the posets based on the topological growth of their distributive lattices, offering a new approach to constructive enumeration of poset families.
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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 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.004 | 0.007 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.013 | 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".