Grove calculator

def Grove(G,parti):
    grove=0
    v_partitions=SetPartitions(Set(G.vertices())).list()
    for v_parti in v_partitions:
        count=0
        forest_poly=1
        if len(v_parti)==len(parti):
            v_parti_list=list(v_parti)[:]
            for i in parti:
                subcount=0
                for j in v_parti_list:
                    if len(j)>0:
                        j=set(list(j))
                    else:
                        j={}
                    if i in Subsets(j).list():
                        #print(i,'in',j,type(j),v_parti_list)
                        subcount+=1
                        if j in v_parti_list:
                            v_parti_list.remove(j)
                if subcount>0:
                    count+=1         
            if count==len(parti):
                #print(parti,'and',v_parti)
                for part in v_parti:
                    if len(part)>1:
                        tree_poly=0
                        subG=G.subgraph(list(part))
                        trees=subG.spanning_trees()
                        for tree in trees:
                            sub_tree_poly=1
                            for eg in tree.edges():
                                sub_tree_poly*=G.edge_label(eg[0],eg[1])
                            tree_poly+=sub_tree_poly
                    else:
                        tree_poly=1
                    forest_poly*=tree_poly
            else:
                forest_poly=0
        else:
            forest_poly=0 
        grove+=forest_poly
    return grove

Make your graph here

def make_graph():
    E=[(4,1,'a'),(4,2,'b'),(4,3,'c'),(1,3,'e'),(1,2,'f'),(2,3,'g')]
    return E

def Graff(E):
    G=[]
    for i in E:
        G+=[(i[0],i[1],var(i[2]))]
    graph=Graph(G)
    return graph

myGraph=Graff(make_graph())

myGraph.graphplot(edge_labels=True,layout='planar').show()

Some examples

Grove(myGraph,[{1,2,3},{4}])

e*f + e*g + f*g

Grove(myGraph,[{1},{2},{3}])

a + b + c

Grove(myGraph,[{1,2},{3,4}])

c*f

Grove(myGraph,[{1},{2}])

a*c + b*c + a*e + b*e + c*e + a*g + b*g + c*g

Grove(myGraph,[{1},{2,3}])

b*c + a*g + b*g + c*g

Grove(myGraph,[{1},{3,4}])

b*c + c*f + b*g + c*g

Grove(myGraph,[{1}])

a*b*c + a*b*e + b*c*e + a*c*f + b*c*f + a*e*f + b*e*f + c*e*f + a*b*g + a*c*g + a*e*g + b*e*g + c*e*g + a*f*g + b*f*g + c*f*g