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Finiteness theorems for Cayley permutation classes
Abstract
We prove that every hereditary Cayley permutation class with finitely many simple members has a finite basis and an algebraic generating function. We first show that every simple Cayley permutation of length at least two contains a simple pattern obtained by deleting one or two letters, and that the only exceptional Cayley permutations are the simple parallel alternations and their symmetries. We then characterize substitution-closed classes by the simplicity of their basis elements and show that classes with finitely many simple members are partially well ordered. Given the simple members and the basis of such a class, we construct a finite system of functional equations and compute an annihilating polynomial.
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BibTeX
@misc{cerbai2026finiteness,
title = {{Finiteness theorems for Cayley permutation classes}},
author = {Giulio Cerbai and Anders Claesson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30484},
eprint = {2610.30484},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30484},
}
RIS
TY - UNPB
TI - Finiteness theorems for Cayley permutation classes
AU - Giulio Cerbai
AU - Anders Claesson
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.30484
UR - https://arxiv.org/abs/2610.30484
AB - We prove that every hereditary Cayley permutation class with finitely many simple members has a finite basis and an algebraic generating function. We first show that every simple Cayley permutation of length at least two contains a simple pattern obtained by deleting one or two letters, and that the only exceptional Cayley permutations are the simple parallel alternations and their symmetries. We then characterize substitution-closed classes by the simplicity of their basis elements and show that classes with finitely many simple members are partially well ordered. Given the simple members and the basis of such a class, we construct a finite system of functional equations and compute an annihilating polynomial.
ER -