Counting connected partitions of graphs

Motivated by the theorem of Győri and Lovász, we consider the following problem. For a connected graph $G$ on $n$ vertices and $m$ edges determine the number $P\left(G,k\right)$ of unordered solutions of positive integers ${\sum }_{i=1}^k{m}_i=m$ such that every $m_i$ is realized by a connected subgraph $H_i$ of $G$ with $m_i$ edges. We also consider the vertex-partition analogue. We prove various lower bounds on $P\left(G,k\right)$ as a function of the number $n$ of vertices in $G$, as a function of the average degree $d$ of $G$, and also as the size $CM{C}_r\left(G\right)$ of $r$-partite connected maximum cuts of $G$. Those three lower bounds are tight up to a multiplicative constant. We also prove that the number $\pi \left(G,k\right)$ of unordered $k$-tuples with ${\sum }_{i=1}^k{n}_i=n$, that are realizable by vertex partitions into $k$ connected parts, is at least $\Omega \left(d^{k-1}\right)$.