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S-meandric Permutations and Tangency Polynomials

  • Yury Belousov
  • Viktoriia Georgievskaia
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Abstract

A meander is a configuration of two simple plane curves intersecting transversely. The orders of their intersection points define a permutation that determines the configuration. When tangencies are allowed, however, different configurations can share the same permutation. We study the combinatorial and algebraic structures arising from this non-uniqueness. We give a realization criterion and show that the realizations of each realizable permutation form an affine space over the two-element field. We describe this space using an associated graph, called the component spine. We prove that the component spine of every permutation is a cactus. We also introduce the tangency polynomial, which counts realizations by their number of tangencies, investigate its properties, and prove that it factors over the cycles and bridges of the component spine. We derive a central limit theorem for tangency counts and obtain asymptotic formulas for the number of distinct tangency polynomials.

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Cite

BibTeXRIS
BibTeX
@misc{belousov2026smeandric,
  title = {{S-meandric Permutations and Tangency Polynomials}},
  author = {Yury Belousov and Viktoriia Georgievskaia},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30068},
  eprint = {2609.30068},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30068},
}
RIS
TY  - UNPB
TI  - S-meandric Permutations and Tangency Polynomials
AU  - Yury Belousov
AU  - Viktoriia Georgievskaia
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.30068
UR  - https://arxiv.org/abs/2609.30068
AB  - A meander is a configuration of two simple plane curves intersecting transversely. The orders of their intersection points define a permutation that determines the configuration. When tangencies are allowed, however, different configurations can share the same permutation. We study the combinatorial and algebraic structures arising from this non-uniqueness. We give a realization criterion and show that the realizations of each realizable permutation form an affine space over the two-element field. We describe this space using an associated graph, called the component spine. We prove that the component spine of every permutation is a cactus. We also introduce the tangency polynomial, which counts realizations by their number of tangencies, investigate its properties, and prove that it factors over the cycles and bridges of the component spine. We derive a central limit theorem for tangency counts and obtain asymptotic formulas for the number of distinct tangency polynomials.
ER  -