A family of invariants of rooted forests

Let A be a commutative k-algebra over a field of k and Ξ a linear operator defined on A. We define a family of A-valued invariants Ψ for finite rooted forests by a recurrent algorithm using the operator Ξ and show that the invariant Ψ distinguishes rooted forests if (and only if) it distinguishes rooted trees T, and if (and only if) it is finer than the quantity α(T)=Aut(T) of rooted trees T. We also consider the generating function U(q)=∑n=1∞Unqn with , where is the set of rooted trees with n vertices. We show that the generating function U(q) satisfies the equation Ξ expU(q)=q−1U(q). Consequently, we get a recurrent formula for Un (n 1), namely, U1=Ξ(1) and Un=ΞSn−1(U1,U2,…,Un−1) for any n 2, where 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 Ψ and derive some consequences about these well-known invariants from our general results on Ψ. Finally, we generalize the invariant Ψ to labeled planar forests and discuss its certain relations with the Hopf algebra in Foissy (Bull. Sci. Math. 126 (2002) 193) spanned by labeled planar forests.