The notation of a partition of type $n$, $(n\in N)$, was introduced by J. Hartmanis in [1] as a generalization of the notion of an ordinary partition of a set. It is well-known fact that partitions $Q$ (of type 1) correspond in an one-one way to equivalence relations on $Q$. In this article we introduce an analogous family of relations $(\mathcal{F}_{n}(Q))$ for partitions of type $n$. Furthermore for $p\in \mathcal{F}_{n}(Q)$ the following statements hold: $○(\overset{n+1}{\rho}) = \rho$ and (verset{n}{\rho})^{-1} = \rho$ for $n = 1: im ○ im = im$ and $im$ (cf. [3]). A similar family of relations for partitions of type $n$ was described by H.E. Pickett in [2] point out the differences.