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PreprintCC BY 4.0arXiv

Iterating the Lehmer code on inversion sequences: Catalan fixed points and finite stabilization

  • Julian Allagan
  • Shanzhen Gao
  • Benjamin Testart
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Abstract

We study an operator Θ on finite integer sequences, where Θ(σ)i counts the entries to the left of σi that are strictly smaller than σi. This operator is a variant of the so-called Lehmer code. For every sequence σ, the image Θ(σ) is an inversion sequence, and the restriction of Θ to permutations of [0,n−1] is a bijection onto inversion sequences of length n. We characterize the fixed points of Θ by avoidance of the pattern $101$ together with a saturation condition, prove that they are counted by the Catalan numbers, and give an explicit recursive bijection with Dyck paths. We also show that the sequences whose first Θ-image is fixed are precisely those avoiding both $101$ and $201$. Finally, we prove finite stabilization for all inversion sequences, exhibit a family attaining the maximal stabilization time, and show that the second stabilization level is not closed under classical patterns.

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Cite

BibTeXRIS
BibTeX
@misc{allagan2026iterating,
  title = {{Iterating the Lehmer code on inversion sequences: Catalan fixed points and finite stabilization}},
  author = {Julian Allagan and Shanzhen Gao and Benjamin Testart},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.24476},
  eprint = {2608.24476},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.24476},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Iterating the Lehmer code on inversion sequences: Catalan fixed points and finite stabilization
AU  - Julian Allagan
AU  - Shanzhen Gao
AU  - Benjamin Testart
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2608.24476
UR  - https://arxiv.org/abs/2608.24476
AB  - We study an operator Θ on finite integer sequences, where Θ(σ)_i counts the entries to the left of σ_i that are strictly smaller than σ_i. This operator is a variant of the so-called Lehmer code. For every sequence σ, the image Θ(σ) is an inversion sequence, and the restriction of Θ to permutations of [0,n−1] is a bijection onto inversion sequences of length n. We characterize the fixed points of Θ by avoidance of the pattern $101$ together with a saturation condition, prove that they are counted by the Catalan numbers, and give an explicit recursive bijection with Dyck paths. We also show that the sequences whose first Θ-image is fixed are precisely those avoiding both $101$ and $201$. Finally, we prove finite stabilization for all inversion sequences, exhibit a family attaining the maximal stabilization time, and show that the second stabilization level is not closed under classical patterns.
ER  -