Shuffle automata and the growth of 1324-avoiding permutations

@misc{brignall2026shuffle,
  title = {{Shuffle automata and the growth of 1324-avoiding permutations}},
  author = {Robert Brignall},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02535},
  eprint = {2610.02535},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02535},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Shuffle automata and the growth of 1324-avoiding permutations
AU  - Robert Brignall
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.02535
UR  - https://arxiv.org/abs/2610.02535
AB  - We show that gr(Av(1324))≤ 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.
ER  -