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

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