Principal specializations of the t-chromatic symmetric function
Abstract
We introduce the t-chromatic symmetric function ZG(x;t), a variant of Stanley's chromatic symmetric function in which the variable t records the number of distinct colors used in a proper coloring. Although ZG(x;t) is equivalent to the chromatic symmetric function as a graph invariant, its principal specializations retain substantially more information than those of the chromatic symmetric function. This construction is motivated by the work of Bajo--Beck--Vindas-Meléndez on the q-chromatic polynomial and the conjecture of Loehr--Warrington that the nth principal specialization of the chromatic symmetric function distinguishes trees on at most n vertices, both of which stem from Stanley's longstanding question of whether the chromatic symmetric function distinguishes trees. We determine the classes of caterpillars resulting from the second principal specialization of ZG(x;t), showing that the second specialization already captures some of the structure of trees. We then prove that the third principal specialization distinguishes and explicitly reconstructs caterpillars with three spine vertices as well as elongated bistars. Finally, we conjecture, based on computational data, that the fourth principal specialization distinguishes all caterpillars.
Related
- The chromatic noncommutative symmetric function of oriented trees
- Chromatic symmetric functions for annular webs
- First-Derivative Chromatic Symmetric Reconstruction For Proper Trees
- Reduction of symmetric time-dependent Hamiltonian systems I: presymplectic principal ℝ-bundles
- A generating function identity for finite and symmetric multiple zeta values via Dougall's bilateral 2H2 summation
BibTeX
@misc{bingham2026principal,
title = {{Principal specializations of the $t$-chromatic symmetric function}},
author = {Aram Bingham and Kyle Celano and Patricia Commins and Rosa Orellana and Jianping Pan and Mary Claire Simone},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31785},
eprint = {2610.31785},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31785},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Principal specializations of the t-chromatic symmetric function
AU - Aram Bingham
AU - Kyle Celano
AU - Patricia Commins
AU - Rosa Orellana
AU - Jianping Pan
AU - Mary Claire Simone
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31785
UR - https://arxiv.org/abs/2610.31785
AB - We introduce the t-chromatic symmetric function Z_G(x;t), a variant of Stanley's chromatic symmetric function in which the variable t records the number of distinct colors used in a proper coloring. Although Z_G(x;t) is equivalent to the chromatic symmetric function as a graph invariant, its principal specializations retain substantially more information than those of the chromatic symmetric function. This construction is motivated by the work of Bajo--Beck--Vindas-Meléndez on the q-chromatic polynomial and the conjecture of Loehr--Warrington that the nth principal specialization of the chromatic symmetric function distinguishes trees on at most n vertices, both of which stem from Stanley's longstanding question of whether the chromatic symmetric function distinguishes trees. We determine the classes of caterpillars resulting from the second principal specialization of Z_G(x;t), showing that the second specialization already captures some of the structure of trees. We then prove that the third principal specialization distinguishes and explicitly reconstructs caterpillars with three spine vertices as well as elongated bistars. Finally, we conjecture, based on computational data, that the fourth principal specialization distinguishes all caterpillars.
ER -