We study the symmetric function and polynomial combinatorial invariants of Hopf algebras of permutations, posets and graphs. We investigate their properties and the relations among them. In particular, we show that the chromatic symmetric function and many other invariants have a property we call positively $h$-alternating. This property of positively $h$-alternating leads to Schur positivity and $e$-positivity when applying the operator $\nabla$ at $q=1$. We conclude by showing that the invariants we consider can be expressed as scheduling problems.

中文翻译：

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排列、顺序和图形的新不变量

我们研究置换、偏序集和图的 Hopf 代数的对称函数和多项式组合不变量。我们研究它们的特性和它们之间的关系。特别是，我们证明了色对称函数和许多其他不变量具有我们称之为正 $h$-交替的属性。当在 $q=1$ 处应用运算符 $\nabla$ 时，正 $h$-交替的这种性质导致 Schur 正性和 $e$-正性。最后，我们证明我们考虑的不变量可以表示为调度问题。