A Family of Invariants of Rooted Forests

Let $A$ be a commutative $k$-algebra over a field of $k$ and $\Xi$ a linear operator defined on $A$. We define a family of $A$-valued invariants $\Psi$ for finite rooted forests by a recurrent algorithm using the operator $\Xi$ and show that the invariant $\Psi$ distinguishes rooted forests if (and only if) it distinguishes rooted trees $T$, and if (and only if) it is {\it finer} than the quantity $\alpha (T)=|\text{Aut}(T)|$ of rooted trees $T$. We also consider the generating function $U(q)=\sum_{n=1}^\infty U_n q^n$ with $U_n =\sum_{T\in \bT_n} \frac 1{\alpha (T)} \Psi (T)$, where $\bT_n$ is the set of rooted trees with $n$ vertices. We show that the generating function $U(q)$ satisfies the equation $\Xi \exp U(q)= q^{-1} U(q)$. Consequently, we get a recurrent formula for $U_n$ $(n\geq 1)$, namely, $U_1=\Xi(1)$ and $U_n =\Xi S_{n-1}(U_1, U_2, >..., U_{n-1})$ for any $n\geq 2$, where $S_n(x_1, x_2, ...)$ $(n\in \bN)$ are the elementary Schur polynomials. We also show that the (strict) order polynomials and two well known quasi-symmetric function invariants of rooted forests are in the family of invariants $\Psi$ and derive some consequences about these well-known invariants from our general results on $\Psi$. Finally, we generalize the invariant $\Psi$ to labeled planar forests and discuss its certain relations with the Hopf algebra $\mathcal H_{P, R}^D$ in \cite{F} spanned by labeled planar forests.