# Tutte polynomial from independent sets of a graph

Let $G$ be a connected graph with chromatic polynomial $X(G,q)$. Since $k$-proper coloring a graph is same as partitioning the vertex set $V$ into $k$ independent sets (a subset of the vertex set in which no two vertices are adjacent) (we call this $k$-independent partition), the chromatic polynomial can be retrieved if we know all the independent subsets of $V$. For example, we have the following expression:

$$X(G,q) = \sum_{k \ge 1} \binom{q}{k} C_k(G)$$ where $C_k(G)$ is the number of number of $k$-independent partitions of $V$.

My question is, is there such a nice expression (fully in terms of independent sets) for Tutte polynomial?