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First-Derivative Chromatic Symmetric Reconstruction For Proper Trees
Abstract
Richard Stanley asked whether every tree T is determined up to isomorphism by its chromatic symmetric function XT. We approach this open problem by regarding XT as a polynomial in the power-sum symmetric functions p1, p2, … and studying the invariant ΦT = .(∂ XT)/(∂ p1)rvertp1 = 0. We prove that ΦT distinguishes, among proper trees, every proper tree in which any non-leaf vertex with two or more non-leaf neighbors has a number of leaf neighbors occurring at no other non-leaf vertex. We give an equivalent formulation of Stanley's question in terms of ΦT. Finally, we show that counts of spanning forests with at most t edges cannot distinguish all k-vertex trees when t<⌊ k/2⌋, for every k≥4.
Related
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- On Macdonald expansions of 𝑞-chromatic symmetric functions and the Stanley–Stembridge conjecture
- Symmetric Hermite quadrature-based balanced truncation for learning linear dynamical systems from derivative data
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BibTeX
@misc{awan2026firstderivative,
title = {{First-Derivative Chromatic Symmetric Reconstruction For Proper Trees}},
author = {Saad A. Awan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.13464},
eprint = {2609.13464},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13464},
}
RIS
TY - UNPB
TI - First-Derivative Chromatic Symmetric Reconstruction For Proper Trees
AU - Saad A. Awan
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2609.13464
UR - https://arxiv.org/abs/2609.13464
AB - Richard Stanley asked whether every tree T is determined up to isomorphism by its chromatic symmetric function X_T. We approach this open problem by regarding X_T as a polynomial in the power-sum symmetric functions p_1, p_2, … and studying the invariant Φ_T = .(∂ X_T)/(∂ p_1)rvert_p_1 = 0. We prove that Φ_T distinguishes, among proper trees, every proper tree in which any non-leaf vertex with two or more non-leaf neighbors has a number of leaf neighbors occurring at no other non-leaf vertex. We give an equivalent formulation of Stanley's question in terms of Φ_T. Finally, we show that counts of spanning forests with at most t edges cannot distinguish all k-vertex trees when t<⌊ k/2⌋, for every k≥4.
ER -