真圆弧图是 $e$-正的

中文摘要 AI 辅助

我们证明了真圆弧图的色对称函数的一个 $e$-正公式，解决了 Ellzey 猜想的 $q=1$ 情形。由此，我们给出了单位区间图 $e$-正性的一个新证明，该证明与 Guay-Paquet 的约化相结合，为 Stanley--Stembridge 猜想提供了一个新证明。我们定义了计数正常染色的颜色矩阵，以及表矩阵，其元素是非负有理数和初等对称函数的比值。我们证明了这两个矩阵通过一族基变换矩阵相关联，这些基变换矩阵在限制到有限多种颜色后变为可逆的，并且我们证明了真圆弧图的色对称函数来自这些矩阵的迹。利用 Hikita 的表，这给出了真圆弧图的色对称函数的一个显式公式，即对前 $k$ 个和后 $k$ 个顶点位于同一列的表上的加权和。

英文摘要

We prove an $e$-positive formula for the chromatic symmetric function of proper circular arc graphs solving the $q=1$ case of Ellzey's conjecture. In doing so, we provide a new proof of the $e$-positivity of unit interval graphs, which alongside Guay-Paquet's reduction gives a new proof of the Stanley--Stembridge conjecture. We define color matrices, which count proper colorings, and tableau matrices, whose entries are nonnegative rational numbers and ratios of elementary symmetric functions. We prove the two matrices are related by a single family of change of basis matrices, which become invertible after restricting to finitely many colors, and we show the chromatic symmetric function of proper circular arc graphs comes from taking the trace of these matrices. Using Hikita's tableaux, this gives an explicit formula for the chromatic symmetric function of a proper circular arc graph as a weighted sum over tableaux whose first and last $k$ vertices lie in the same columns.

发表机构

• Department of Mathematics, Massachusetts Institute of Technology（麻省理工学院数学系）

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