A ChatGPT-assisted Triangle Characterization of Affine Permutation Inversion Graphs

Abstract

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.