Permutation groups, pattern involvement, and Galois connections

Erkko Lehtonen; Reinhard Pöschel

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arXiv · 1605.04516

Permutation groups, pattern involvement, and Galois connections

Erkko Lehtonen·Reinhard Pöschel

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Abstract

There is a connection between permutation groups and permutation patterns: for any subgroup $G$ of the symmetric group $S_\ell$ and for any $n \geq \ell$, the set of $n$-permutations involving only members of $G$ as $\ell$-patterns is a subgroup of $S_n$. Making use of the monotone Galois connection induced by the pattern avoidance relation, we characterize the permutation groups that arise via pattern avoidance as automorphism groups of relations of a certain special form. We also investigate a related monotone Galois connection for permutation groups and describe its closed sets and kernels as automorphism groups of relations.

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Erkko Lehtonen, Reinhard Pöschel. 2017-02-01. Permutation groups, pattern involvement, and Galois connections. https://arxiv.org/abs/1605.04516

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