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On the generation of the symmetric group by index relations of three prefix reversals

  • Yexiao Lei
  • Jinxing Zhao
  • Ziyu Zhu
  • Yang Wu
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Abstract

The cubic pancake graphs are Cayley graphs over the symmetric group Sn generated by three prefix reversals. There is the folllowing open problem: characterize all the sets of three prefix reversals that generate Sn. Blanco and Golubyatnikov give a complete solution of the problem when any of the two smallest or the two largest lengths but n are included in a triple of prefix reversals. According to their conjectures, we give some conditions implying a triple of prefix reversals does not generate Sn. Moreover, Using the symmetric groups that can already be generated by three prefix reversals, we can derive some recurrence conditions.

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Cite

BibTeXRIS
BibTeX
@misc{lei2026generation,
  title = {{On the generation of the symmetric group by index relations of three prefix reversals}},
  author = {Yexiao Lei and Jinxing Zhao and Ziyu Zhu and Yang Wu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30892},
  eprint = {2610.30892},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30892},
}
RIS
TY  - UNPB
TI  - On the generation of the symmetric group by index relations of three prefix reversals
AU  - Yexiao Lei
AU  - Jinxing Zhao
AU  - Ziyu Zhu
AU  - Yang Wu
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.30892
UR  - https://arxiv.org/abs/2610.30892
AB  - The cubic pancake graphs are Cayley graphs over the symmetric group Sn generated by three prefix reversals. There is the folllowing open problem: characterize all the sets of three prefix reversals that generate Sn. Blanco and Golubyatnikov give a complete solution of the problem when any of the two smallest or the two largest lengths but n are included in a triple of prefix reversals. According to their conjectures, we give some conditions implying a triple of prefix reversals does not generate Sn. Moreover, Using the symmetric groups that can already be generated by three prefix reversals, we can derive some recurrence conditions.
ER  -