Bibliographic record
Abstract
The chromatic symmetric function X G is a symmetric function generalization of the chromatic polynomial of a graph, introduced by Stanley [7] . Stanley [7] gave an expansion formula for X G in terms of the power sum symmetric functions p λ using the principle of inclusion-exclusion, and Bernardi and Nadeau [1] gave an alternate p -expansion for X G in terms of acyclic orientations. Crew, Pechenik, and Spirkl [3] defined the Kromatic symmetric function X ‾ G as a K -theoretic analogue of X G , constructed in the same way except that each vertex is assigned a nonempty set of colors such that adjacent vertices have nonoverlapping color sets. They defined a K -analogue p ‾ λ of the power sum basis and computed the first few coefficients of the p ‾ -expansion of X ‾ G for some small graphs G . They conjectured that the p ‾ -expansion always has integer coefficients and asked whether there is an explicit formula for these coefficients. In this note, we give a formula for the p ‾ -expansion of X ‾ G , show two ways to compute the coefficients recursively (along with examples), and prove that the coefficients are indeed always integers. In a more recent paper [6] , we use our formula from this note to give a combinatorial description of the p ‾ -coefficients [ p ‾ λ ] X ‾ G and a simple characterization of their signs in the case of unweighted graphs.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 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".