Generalized Bijective Maps between $G$-Parking Functions, Spanning Trees, and the Tutte Polynomial

Abstract

We introduce an object called a tree growing sequence (TGS) in an effort to generalize bijective correspondences between $G$-parking functions, spanning trees, and the set of monomials in the Tutte polynomial of a graph $G$. A tree growing sequence determines an algorithm which can be applied to a single function, or to the set $\mathcal{P}_{G,q}$ of $G$-parking functions. When the latter is chosen, the algorithm uses splitting operations - inspired by the recursive defintion of the Tutte polynomial - to iteratively break $\mathcal{P}_{G,q}$ into disjoint subsets. This results in bijective maps $\tau$ and $\rho$ from $\mathcal{P}_{G,q}$ to the spanning trees of $G$ and Tutte monomials, respectively. We compare the TGS algorithm to Dhar's algorithm and the family described by Chebikin and Pylyavskyy. Finally, we compute a Tutte polynomial of a zonotopal tiling using analogous splitting operations.