  For a mapping class on a punctured surface, we prove that the pseudo-Anosov
property is equivalent to the sign stability introduced in [IK19]. In
particular we conclude that the algebraic entropies of the $\mathcal{A}$- and
$\mathcal{X}$-cluster transformations induced by a mutation loop given by a
pseudo-Anosov mapping class both coincide with its topological entropy.
