The partitions whose members are finite and the permutations with at most \(n\) non-fixed points of a set

We write \(S_{\leq n}(A)\) and \(\Part_{\fin}(A)\) for the set of permutations with at most \(n\) non-fixed points, where \(n\) is a natural number, and the set of partitions whose members are finite, respectively, of a set \(A\). Among our results, we show, in the Zermelo-Fraenkel set theory, that...

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Bibliographic Details
Published in:arXiv.org 2023-12
Main Authors: Sonpanow, Nattapon, Vejjajiva, Pimpen
Format: Article
Language:English
Subjects:
Partitions (mathematics)
Permutations
Set theory
Online Access:Get full text
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Summary:We write \(S_{\leq n}(A)\) and \(\Part_{\fin}(A)\) for the set of permutations with at most \(n\) non-fixed points, where \(n\) is a natural number, and the set of partitions whose members are finite, respectively, of a set \(A\). Among our results, we show, in the Zermelo-Fraenkel set theory, that \(|\Part_{\fin}(A)| \nleq |S_{\leq n}(A)|\) for any infinite set \(A\) and if \(A\) can be linearly ordered, then \(|S_{\leq n}(A)| < |\Part_{\fin}(A)|\) while the statement ``\(|S_{\leq n}(A)|\leq|\Part_{\fin}(A)|\) for all infinite sets \(A\)" is not provable for \(n\geq 3\).
ISSN:2331-8422