  We estimate size of recurrence of an action of a nilpotent group by
homeomorphisms of a compact space for polynomial mappings into a nilpotent
group form the partial semigroup $(\mathcal{P}_{f}(\mathbb{N}),\uplus)$. To do
this we have used algebraic structure of the Stone-\v{C}ech copactification
partial semigroup and that of the given nilpotent group.
