A quantitative way to e-positivity of trees

In 2020, Dahlberg, She, and van Willigenburg conjectured that the chromatic symmetric function of any tree with maximum degree at least 4 is not e-positive. Zheng and Tom verified this conjecture for all trees with maximum degree at least 5 and spiders with maximum degree 4, and in their proofs the following necessary condition given by Wolfgang plays an important role: every connected graph having e-positive chromatic symmetric function must contain a connected partition of every type. In order to make further progress on this conjecture, we refine Wolfgang’s result in a quantitative way. At first, we give an explicit formula for the e-coefficients of trees in terms of their connected partitions, by which e-positivity is equivalent to a series of inequalities for the numbers of connected partitions. Based on this formula, we present several necessary conditions on the numbers of connected partitions or acyclic orientations for trees to be e-positive. These necessary conditions turn out to be characterizations on the structure of e-positive trees, and as sample applications we prove the non-e-positivity of several classes of trees with maximum degree 3 or 4. We further make more discussions and calculations on trees with maximum degree 4 and having a connected partition of every type, which inspire us to come up with a list of open problems towards the final resolution of the above conjecture.

💡 Research Summary

The paper addresses the long‑standing conjecture that any tree whose maximum degree is at least four cannot be e‑positive, i.e. its chromatic symmetric function cannot be expressed as a non‑negative linear combination of elementary symmetric functions. Building on a qualitative result of Wolfgang—who proved that an e‑positive connected graph must admit a connected partition of every integer partition type—the authors develop a quantitative framework that translates e‑positivity into explicit inequalities involving the numbers of connected partitions of a tree.

The authors first recall the necessary background on symmetric functions, introducing elementary symmetric functions (e_{\lambda}) and power‑sum symmetric functions (p_{\lambda}). They use the notion of brick tabloids, originally due to Gessel, to describe the transition matrix between the (e)‑basis and the (p)‑basis. For a partition (\lambda) they denote by (B_{\lambda,\mu}) the set of brick tabloids of shape (\mu) and content (\lambda), and by (w(B_{\lambda,\mu})) the sum of the products of brick lengths. The key identities (2.1) and (2.2) express (e_{\mu}) and (p_{\mu}) in terms of each other via these objects.

Using Stanley’s expansion of the chromatic symmetric function in the power‑sum basis, the authors obtain for any tree (T) the formula \