发表机构
School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
解决Shelburne和van Willigenburg留下的关于图\(G = K_{(3,\,2^\beta)}\)是否为\(e\)-正的问题，通过推导标记变量系数提取公式，结合迪克森多项式等，刻画所有\(e\)-正的完全多部图。
AI 中文摘要
Shelburne和van Willigenburg（arXiv:2604.26158）刻画了舒尔正的完全多部图，并留下了图\(G = K_{(3,\,2^\beta)}\)是否为\(e\)-正的问题。我们解决了这个问题，并与他们的分类一起，刻画了所有\(e\)-正的完全多部图。我们的主要结果是\(X_G\)的一个显式的、明显非负的\(e\)-展开式，其系数用受限注入数表示。我们的主要思想是从基本单项式柯西恒等式推导出任意完全多部图的\(e\)-系数的标记变量系数提取公式。对于特定的图\(G\)，该公式将证明简化为三个系数族，我们使用迪克森多项式和这些数的递推关系对其进行求值。
英文摘要
Shelburne and van Willigenburg (arXiv:2604.26158) characterize the Schur-positive complete multipartite graphs and leave open whether the graphs~$G=K_{(3,\,2^β)}$ are $e$-positive. We resolve this question and, together with their classification, characterize all $e$-positive complete multipartite graphs. Our main result is an explicit, manifestly nonnegative $e$-expansion of~$X_G$ whose coefficients are expressed in terms of the restricted-injection numbers. Our main idea is to derive a marker-variable coefficient-extraction formula for the $e$-coefficients of arbitrary complete multipartite graphs from the elementary--monomial Cauchy identity. For the particular graph~$G$, this formula reduces the proof to three coefficient families, which we evaluate using Dickson polynomials and recurrences for these numbers.
