AI 中文总结
研究借助ChatGPT给出仿射置换逆序图的新刻画，基于加权竞赛图三角形局部条件，带来识别逆序图和集的高效算法、有向路径权重界等，还给出新算法测试集合是否为仿射置换逆序集。
AI 中文摘要
Björner和Brenti广泛研究了仿射对称群$\widetilde{S}_n$中置换的逆序集。他们编码逆序集的一种方法是通过仿射逆序图，它是顶点集为$[n]=\{1,2,\ldots,n\}$的特定加权图。Papi随后的工作刻画了哪些图可作为仿射逆序图出现。本文借助ChatGPT提供了一种基于加权竞赛图中每个三角形的简单局部条件的替代刻画。新刻画带来了识别逆序图和逆序集的高效算法，还给出了有向路径上权重的界等。最后给出了一个新的$O(|R|+n^{3})$算法来测试给定集合$R$是否为仿射置换的逆序集。
英文摘要
Inversion sets of permutations in the affine symmetric group $\widetilde{S}_n$ were studied extensively by Björner and Brenti. One of their methods for encoding an inversion set is through an affine inversion graph, which is a certain weighted graph on vertex set $[n]=\{1,2,\ldots,n\}$. Subsequent work by Papi characterized which graphs arise as affine inversion graphs. In this paper, we provide an alternative characterization in terms of a simple local condition on each triangle in a weighted tournament graph. This new characterization was produced with the assistance of ChatGPT, which suggested several key insights that simplified portions of Papi's original characterization. Consequences of our characterization include efficient algorithms for recognizing inversion graphs and inversion sets. Furthermore, we give bounds on the weights along directed paths, and we show that standardizing the labels on an induced subgraph results in another inversion graph. We conclude with a new order $O(|R|+n^{3})$ algorithm for testing if a given set $R$ is the inversion set of an affine permutation.
Comments13 pages, 2 figures
