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Mathematics > Combinatorics

arXiv:2001.00181 (math)
[Submitted on 1 Jan 2020 (v1), last revised 16 Jan 2020 (this version, v2)]

Title:Non-Schur-positivity of chromatic symmetric functions

Authors:David G.L. Wang, Monica M.Y. Wang
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Abstract:We provide a formula for every Schur coefficient in the chromatic symmetric function of a graph in terms of special rim hook tabloids. This formula is useful in confirming the non-Schur positivity of the chromatic symmetric function of a graph, especially when Stanley's stable partition method does not work. As applications, we determine Schur positive fan graphs and Schur positive complete tripartite graphs. We show that any squid graph obtained by adding $n$ leaves to a common vertex on an $m$-vertex cycle is not Schur positive if $m\ne 2n-1$, and conjecture that neither are the squid graphs with $m=2n-1$.
Comments: 12 pages, 5 figures
Subjects: Combinatorics (math.CO)
MSC classes: 05E05 05A17 05A15 05C70
Cite as: arXiv:2001.00181 [math.CO]
  (or arXiv:2001.00181v2 [math.CO] for this version)
  https://doi.org/10.48550/arXiv.2001.00181

Submission history

From: David Wang [view email]
[v1] Wed, 1 Jan 2020 10:13:06 UTC (19 KB)
[v2] Thu, 16 Jan 2020 17:38:33 UTC (18 KB)
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