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

In this paper, we investigate relationships between |seq(A)|and|Part(fin)(A)|in the absence of the Axiom of Choice, where seq(A)is the set of finite sequences of elements in a set A and Part(fin)(A)is the set of partitions of A whose members are finite. We show that |seq(A)|<|Part(fin)(A)|if A is Dedekind-infinite and the condition cannot be removed. Moreover, this relationship holds for an arbitrary infinite set A if we restrict seq(A)to the set of finite sequences with a bounded length.