In the present paper, we introduce the sequence space \[{l_p}(E,\Delta) = \left\{ x = (x_n)_{n = 1}^\infty : \sum_{n = 1}^\infty \left| \sum_{j \in {E_n}} x_j - \sum_{j \in E_{n + 1}} x_j\right| ^p < \infty \right\},\] where $E=(E_n)$ is a partition of finite subsets of the positive integers and $p\ge 1$. We investigate its topological properties and inclusion relations. Moreover, we consider the problem of finding the norm of certain matrix operators from $l_p$ into $ l_p(E,\Delta)$, and apply our results to Copson and Hilbert matrices.