Year: 2026
Type: preprint
Abstract: We show that $\text{gr}(\text{Av}(1324))\leq 13.167248$. This is achieved by combining the approach of Bevan, Brignall, Elvey Price and Pantone using the `domino', with ideas from earlier upper bounds by Bóna using pairs of decorated words with additional restrictions. More specifically, we replace the arbitrary interleavings of Bevan et. al. with interleavings more like those of Bóna. Our method uses shuffle automata to handle the interleavings, and this method has the potential to provide better upper bounds than the one established here. To estimate how many shuffles are possible, we use two key statistics on dominoes: the number of internal points (that is, points that are neither left-to-right minima not right-to-left maxima), and the number of runs of internal points.
Source: arXiv (Cornell University)
Author: Brignall, Robert
Institution: The Open University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220428433