A quantitative way to e-positivity of trees
Ethan Y. H. Li
TL;DR
This paper provides a quantitative refinement of Wolfgang's result on $e$-positivity for trees by deriving an explicit formula for the $e$-coefficients of a tree $T$ in terms of connected partitions and brick-tabloid weights. The main contribution is the formula $[e_{\,\lambda}] X_T = \sum_{\mu \dashv \lambda} (-1)^{\ell(\lambda)-\ell(\mu)} b_{\mu} w(B_{\lambda,\mu})$, together with derived necessary conditions on the number of connected partitions of various types and on acyclic orientations (sinks) for $e$-positivity. The authors apply these tools to show non-$e$-positivity for several caterpillars and to establish a necessary condition for caterpillars to be $e$-positive, providing supporting evidence for the conjecture that trees with maximum degree at least $4$ are not $e$-positive. The results connect combinatorial partition data with chromatic symmetric function expansions and yield practical tests for $e$-positivity in trees.
Abstract
In 1997, Wolfgang proved that every connected graph having $e$-positive chromatic symmetric function must contain connected partitions of every type. In this paper, we refine this result by a quantitative way in the special case of trees. At first, we give a formula for calculating $e$-coefficients of trees in terms of their connected partitions. Based on this formula, we present several necessary conditions on the number of connected partitions or acyclic orientations for trees to be $e$-positive. As an application, we prove the non-$e$-positivity of a class of caterpillars which have connected partitions of all type. Moreover, we give a necessary condition for caterpillars to be $e$-positive, which may be applied to provide more evidence to the conjecture of Dahlberg, She, and van Willigenburg that every tree of maximum degree at least 4 is non-$e$-positive.