The enumeration of subgroups in finite $p$-groups

We generalize the familiar principle of enumeration due to Hall and establish a new principle for the enumeration of subgroups of any $p$-group $G$ of order $p^m$, based on the following grouptheoretic relation found by the author: $\sum^m_{\lambda=0}(-1)^\lambda p^{\left(\lambda\atop2\right)}\mathscr E_\lambda(G)=0$, where $\mathscr E_\lambda(G)$ is the number of elementary Abelian subgroups of order $p^\lambda$ in $G$.