2020/01/01 by David G. L. Wang, Wang, David G. L., Monica M. Y. Wang +1
Mathematics · #05E05 05A17 05A15 05C70 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05A15 #msc:05A17 #msc:05C70 #msc:05E05
paper · pdf · doi:10.48550/arxiv.2001.00181
12 pages, 5 figures
arxiv created 2020/01/16 · arxiv updated 2020/01/17
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≠ 2n-1, and conjecture that neither are the squid graphs with m=2n-1.