Séminaire Lotharingien de Combinatoire, 93B.118 (2025), 12 pp.
Foster Tom
On e-Positivity of Trees and Connected Partitions
Abstract.
We prove that a tree with a vertex of degree at least five must be missing a connected partition of some type and therefore its chromatic symmetric function cannot be e-positive. We prove that this also holds for a tree with a vertex of degree four as long as it is not adjacent to any leaf. This brings us very close to the conjecture by Dahlberg, She, and van Willigenburg of non-e-positivity for all trees with a vertex of degree at least four. We also prove that spiders with four legs cannot have an e-positive chromatic symmetric function.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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