Foatic Homomesy

@misc{mao2026foatic,
  title = {{Foatic Homomesy}},
  author = {Peter Mao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40882},
  eprint = {2610.40882},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40882},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Foatic Homomesy
AU  - Peter Mao
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40882
UR  - https://arxiv.org/abs/2610.40882
AB  - Using a unified method of identifying a telescoping function for each statistic-action pair, we prove conjectures of La Croix and Roby, and of Sheridan-Rossi that various permutation statistics, such as the number of fixed points, are homomesic under Foatic actions: compositions of the Foata bijection and dihedral symmetries of one-line notation. Underlying these results is a family of bijective identities relating permutation statistics across the Foata bijection; one of them --- an unexpected invariance of the number of right-to-left minima under reverse-complement, proved by an explicit bijection --- may be of independent interest. The telescoping functions also turn out to be shared among some Foatic actions, implying that mixing certain actions still yields a constant average on closed walks.
ER  -