论文标题
明确公式，用于色素对称函数的电子积极性
Explicit formulas for e-positivity of chromatic quasisymmetric functions
论文作者
论文摘要
1993年，斯坦利（Stanley）和斯蒂姆布里奇（Stembridge）猜想任何$（3+1）$ - 免费poset的色度对称功能为$ e $ - 阳性。 Guay-Paquet将猜想减少到$（3+1）$ - 和$（2+2）$ - 免费POSET，也称为自然单位间隔订单。 Shareshian和Wachs定义了色素对称函数，概括了色度对称函数，并猜想任何自然单位间隔订单的色素对称函数均为$ e $ - 阳性和$ e $ - 单次。 对于给定的自然间隔顺序，有一个相应的分区$λ$，我们用$x_λ$表示色素式对称函数。第一作者引入了局部线性关系，以实现彩色准对称函数。在本文中，我们证明了上述局部线性关系的有力概括，称为矩形引理，这也概括了Huh，Nam和Yoo的结果。这样的引理可以应用于描述$ e $ $ potitivity的明确公式，其中$λ$包含在矩形中的$λ$的$ e $ potitivity。我们还建议通过施加矩形引理时$λ$包含$λ$的$ E $ potitivity的一些猜想公式。
In 1993, Stanley and Stembridge conjectured that a chromatic symmetric function of any $(3+1)$-free poset is $e$-positive. Guay-Paquet reduced the conjecture to $(3+1)$- and $(2+2)$-free posets which are also called natural unit interval orders. Shareshian and Wachs defined chromatic quasisymmetric functions, generalizing chromatic symmetric functions, and conjectured that a chromatic quasisymmetric function of any natural unit interval order is $e$-positive and $e$-unimodal. For a given natural interval order, there is a corresponding partition $λ$ and we denote the chromatic quasisymmetric function by $X_λ$. The first author introduced local linear relations for chromatic quasisymmetric functions. In this paper, we prove a powerful generalization of the above-mentioned local linear relations, called a rectangular lemma, which also generalizes the result of Huh,Nam and Yoo. Such a lemma can be applied to describe explicit formulas for $e$-positivity of a chromatic symmetric function $X_λ$ where $λ$ is contained in a rectangle. We also suggest some conjectural formulas for $e$-positivity when $λ$ is not contained in a rectangle by applying the rectangular lemma.
