Combinatorial identities using the matrix tree theorem
Bibliographic record
Abstract
<p>In this paper, we explore some interesting applications of the matrix tree theorem. In particular, we present a combinatorial interpretation of a distribution of <span class="math inline">\((n-1)^{n-1}\)</span>, in the context of uprooted spanning trees of the complete graph <span class="math inline">\(K_{n}\)</span>, which was previously obtained by Chauve–Dulucq–Guibert. Additionally, we establish a combinatorial explanation for the distribution of <span class="math inline">\(m^{n-1}n^{m-1}\)</span>, related to spanning trees of the complete bipartite graph <span class="math inline">\(K_{m,n}\)</span>, which seems new. Furthermore, we extend this study to the graph <span class="math inline">\(K_{n}\setminus \{e_{1,n}\}\)</span>, obtained by deleting an edge from <span class="math inline">\(K_n\)</span>, and derive a new identity for the number of its uprooted spanning trees.</p>
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How this classification was reachedexpand
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) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.002 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".