The permutations with n non‐fixed points and the subsets with n elements of a set

• Published in: Mathematical logic quarterly 2023-08, Vol.69 (3), p.341-346
• Main Authors: Panasawatwong, Supakun, Vejjajiva, Pimpen
• Format: Article
• Language: English
• Subjects: Permutations
• ISSN: 0942-5616; 1521-3870
• DOI: 10.1002/malq.202300005

We write Sn(a)$\mathcal {S}_n(\mathfrak {a})$ and [a]n$[\mathfrak {a}]^n$ for the cardinalities of the set of permutations with n non‐fixed points and the set of subsets with n elements, respectively, of a set which is of cardinality a$\mathfrak {a}$, where n is a natural number greater than 1. With the Axiom of Choice, Sn(a)$\mathcal {S}_n(\mathfrak {a})$ and [a]n$[\mathfrak {a}]^n$ are equal for all infinite cardinals a$\mathfrak {a}$. We show, in ZF, that if AC≤n$\mbox{\textsf {AC}}_{\le n}$ is assumed, then [a]n≤Sn(a)≤[a]n+1$[\mathfrak {a}]^n\le \mathcal {S}_n(\mathfrak {a})\le [\mathfrak {a}]^{n+1}$ for any infinite cardinal a$\mathfrak {a}$. Moreover, the assumption AC≤n$\mbox{\textsf {AC}}_{\le n}$ cannot be removed for n>2$n>2$ and the superscript n+1$n+1$ cannot be replaced by n. We also show under AC≤n$\mbox{\textsf {AC}}_{\le n}$ that for any infinite cardinal a$\mathfrak {a}$, Sn(a)≤[a]n$\mathcal {S}_n(\mathfrak {a})\le [\mathfrak {a}]^n$ implies a$\mathfrak {a}$ is Dedekind‐infinite.