论文标题
在$ e $ $ - 树木和蜘蛛的积极性上
On the $e$-positivity of trees and spiders
论文作者
论文摘要
我们证明，对于任何至少六个顶点的树，其色度对称函数不是$ e $ prositive，也就是说，它不能写成基本对称函数的非负线性组合。这取得了重大进展，朝着最新的达尔伯格（Dahlberg），她和范·威利根堡（Van Willigenburg）的猜想，他们猜想了所有树木至少四个树木至少四个树木的结果。我们还提供了一系列条件，这些条件可以识别何时蜘蛛的色素对称函数，该树是由末端识别的多个路径组成的树，不是$ e $ sostive。这些条件还推广到具有切割顶点的树木和图形。最后，通过应用Orellana和Scott的结果，我们提供了一种方法，可以在蜘蛛的色度对称函数的基本对称函数扩展中计算某些系数，从而导致蜘蛛的进一步$ e $阳性条件。
We prove that for any tree with a vertex of degree at least six, its chromatic symmetric function is not $e$-positive, that is, it cannot be written as a nonnegative linear combination of elementary symmetric functions. This makes significant progress towards a recent conjecture of Dahlberg, She, and van Willigenburg, who conjectured the result for all trees with a vertex of degree at least four. We also provide a series of conditions that can identify when the chromatic symmetric function of a spider, a tree consisting of multiple paths identified at an end, is not $e$-positive. These conditions also generalize to trees and graphs with cut vertices. Finally, by applying a result of Orellana and Scott, we provide a method to inductively calculate certain coefficients in the elementary symmetric function expansion of the chromatic symmetric function of a spider, leading to further $e$-positivity conditions for spiders.
