Lara Pudwell's solution My conjecture is that P(n,i,a)=0 for all i<=n. This is because both (a[i,1]+a[i,2]+ ... +a[i,n])*TW(n,i,a) and (a[1,i]*TW(n,1,a)+a[2,i]*TW(n,2,a)+ ... + a[n,i]*TW(n,n,a)) count the number of directed graphs on n vertices with n edges that: (1) have exactly one directed cycle, and the cycle includes the vertex i and (2) all vertices NOT in the directed cycle are on paths that are directed towards the cycle. (a[i,1]+a[i,2]+ ... +a[i,n])*TW(n,i,a) counts these graphs by taking all rooted trees on n vertices with root i, and adding an edge from i to another vertex to complete the cycle. (a[1,i]*TW(n,1,a)+a[2,i]*TW(n,2,a)+ ... + a[n,i]*TW(n,n,a)) counts these graphs by taking all rooted trees with vertex k and connecting k to vertex i (and doing this for all k<>i). Since both of these expressions count the same thing, their difference is 0. Lara