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A Family of Invariants of Rooted Forests | | Wenhua Zhao
; | | Date: |
5 Nov 2002 | | Subject: | Combinatorics MSC-class: 05C05; 05A15 | math.CO | | Abstract: | 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 $alpha (T)=| ext{Aut}(T)|$ of rooted trees $T$. We also consider the generating function $U(q)=sum_{n=1}^infty U_n q^n$ with $U_n =sum_{Tin T_n} frac 1{alpha (T)} Psi (T)$, where $T_n$ 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 $U_n$ $(ngeq 1)$, namely, $U_1=Xi(1)$ and $U_n =Xi S_{n-1}(U_1, U_2, >..., U_{n-1})$ for any $ngeq 2$, where $S_n(x_1, x_2, ...)$ $(nin N)$ 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. | | Source: | arXiv, math.CO/0211095 | | Services: | Forum | Review | PDF | Favorites | |
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