The finite sequences and the partitions whose members are finite of a set.

This article investigates the cardinality relationship between the set of finite sequences of elements in a set \( A \), denoted \(\text{seq}(A)\), and the set of finite partitions of \( A \), denoted \(\text{Part}_{\text{fin}}(A)\), within Zermelo–Fraenkel set theory without the Axiom of Choice (ZF). It establishes that for any Dedekind-infinite set \( A \), \(|\text{seq}(A)| < |\text{Part}_{\text{fin}}(A)|\), and this condition cannot be omitted. Additionally, for any infinite set \( A \) and any natural number \( n \), the inequality \(|\text{seq}_{\leq n}(A)| < |\text{Part}_{\text{fin}}(A)|\) holds, where \(\text{seq}_{\leq n}(A)\) is the set of finite sequences of length at most \( n \). The paper also presents a consistency result using permutation models to show that the inequality fails without the Dedekind-infinite assumption, leaving open whether \(|\text{seq}(A)| \neq |\text{Part}_{\text{fin}}(A)|\) holds for all infinite sets in ZF.