无爪图的色对称函数不是舒尔正的
Chromatic symmetric functions of claw-free graphs are not Schur positive
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中文总结 AI 辅助
研究无爪图的色对称函数是否为舒尔正，通过给出线图例子表明其色对称函数在舒尔展开有负系数，还给出反例反驳相关猜想，借助ChatGPT-5.6 Sol Pro找到这些例子。
中文摘要 AI 辅助
色对称函数是代数组合学中经过充分研究的对称函数，它推广了图的色多项式。1995年，斯坦利引入了这些对称函数，并猜想它们对于无爪图是舒尔正的。我们给出了线图（因此是无爪图）的例子，其色对称函数在舒尔展开中有负系数。我们还给出了一个反例，反驳了莫尼卡尔2018年的猜想，即在任何有限变量展开时，舒尔正的色对称函数具有饱和牛顿多面体。这两个例子都是使用ChatGPT-5.6 Sol Pro找到的。
英文摘要
Chromatic symmetric functions are well-studied symmetric functions in algebraic combinatorics that generalize chromatic polynomials of graphs. In 1995, Stanley introduced these symmetric functions and conjectured that they are Schur positive for claw-free graphs. We give examples of a line graphs, which are thus claw-free, whose chromatic symmetric function have a negative coefficient in its Schur expansion. We also give a counterexample to the 2018 conjecture of Monical that Schur positive chromatic symmetric functions have saturated Newton polytope when expanded in any finite number of variables. Both of these examples were found using ChatGPT-5.6 Sol Pro.
