发表机构
Tianjin University; Nankai University（天津大学; 南开大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文引入斯特林子集排列与连续下降统计量，解决了高阶斯特林子集三角的组合解释问题，并证明了相关行对数凹性猜想。
AI 中文摘要
Deb 和 Sokal 在研究组合三角的全正性时引入了 $r$ 阶斯特林循环三角与子集三角及其相关的拟欧拉三角。他们给出了循环情形下基于斯特林排列的组合解释，但子集情形仍悬而未决。对于 $r\ge 2$，我们通过引入斯特林子集排列的概念以及一个连续下降统计量解决了该问题。我们还证明了 Deb 和 Sokal 关于高阶斯特林循环三角与子集三角行对数凹性的猜想。我们的对数凹性证明依赖于 Sagan 准则、Dey 的推广以及借助 ChatGPT 5.6 发现的加强的对数凹性不等式。
英文摘要
The $r$th-order Stirling cycle and subset triangles and their associated quasi-Eulerian triangles were introduced by Deb and Sokal in their study of total positivity of combinatorial triangles. They found combinatorial interpretations for the cycle case in terms of Stirling permutations, leaving the subset case open. For $r\ge 2$, we resolve this problem by introducing the notion of Stirling subset permutations along with a consecutive-descent statistic. We also prove the conjectures of Deb and Sokal on the row log-concavity of the higher-order Stirling cycle and subset triangles. Our log-concavity proofs rely on Sagan's criterion, Dey's extension, and strengthened log-concavity inequalities discovered with the assistance of ChatGPT 5.6.
Comments40 pages
