A refinement of numbers for trees and parking functions

Abstract

A $\emph{leader}$ of a tree $T$ is a vertex which has no smaller descendants in $T$. Gessel and Seo showed that $\displaystyle\sum_{T \in T_n}u^{(\# of leaders in T)}c^{(degree of 1 in T)} = u P_{n-1}(1,u,cu),$ which is a generalization of Cayley``s formula, where $T_n$ is the set of trees on $[n]$ and $P_n(a,b,c) = c \displaystyle\prod_{i=1}^{n-1}(ia+(n-i)b+c).$ Using a variation of the Pr$\ddot{u}$fer code which is called an $\em{RP-code}$, we give a simple bijective proof of Gessel and Seo``s formula. A car in a parking function is called $\emph{lucky}$ if it succeeds to park in its preferred space. Gessel and Seo also showed that $\displaystyle\sum_{P \in PF_n} u^{(\# of lucky cars in P)} = P_n(1,u,u),$ but this proof was not combinatorial. We construct the bijection $\varphi$ from forests to parking functions and give a bijective proof of it. We generalize it further using the bijection $\varphi$.