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.