A family of bijections between G-parking functions and spanning trees

For a directed graph G on vertices {0,1,...,n}, a G-parking function is an n-tuple (b_1,...,b_n) of non-negative integers such that, for every non-empty subset U of {1,...,n}, there exists a vertex j in U for which there are more than b_j edges going from j to G-U. We construct a family of bijective maps between the set P_G of G-parking functions and the set T_G of spanning trees of G rooted at 0, thus providing a combinatorial proof of |P_G| = |T_G|.Comment: 11 pages, 4 figures; a family of bijections containing the two original bijections is presented; submitted to J. Combinatorial Theory, Series

Monomization of Power Ideals and Parking Functions

In this note, we find a monomization of a certain power ideal associated to a directed graph. This power ideal has been studied in several settings. The combinatorial method described here extends earlier work of other, and will work on several other types of power ideals, as will appear in later work

A Bijection Between the Recurrent Configurations of a Hereditary Chip-Firing Model and Spanning Trees

Hereditary chip-firing models generalize the Abelian sandpile model and the cluster firing model to an exponential family of games induced by covers of the vertex set. This generalization retains some desirable properties, e.g. stabilization is independent of firings chosen and each chip-firing equivalence class contains a unique recurrent configuration. In this paper we present an explicit bijection between the recurrent configurations of a hereditary chip-firing model on a graph and its spanning trees.Comment: 13 page

Parking functions on toppling matrices

Let $\Delta$ be an integer $n \times n$-matrix which satisfies the conditions: $\det \Delta\neq 0$, $\Delta_{ij}\leq 0\text{ for }i\neq j,$ and there exists a vector ${\bf r}=(r_1,\ldots,r_n)>0$ such that ${\bf r}\Delta \geq 0$. Here the notation ${\bf r}> 0$ means that $r_i>0$ for all $i$, and ${\bf r}\geq {\bf r}'$ means that $r_i\geq r'_i$ for every $i$. Let $\mathscr{R}(\Delta)$ be the set of vectors ${\bf r}$ such that ${\bf r}>0$ and ${\bf r}\Delta\geq 0$. In this paper, $(\Delta,{\bf r})$-parking functions are defined for any ${\bf r}\in\mathscr{R}(\Delta)$. It is proved that the set of $(\Delta,{\bf r})$-parking functions is independent of ${\bf r}$ for any ${\bf r}\in\mathscr{R}(\Delta)$. For this reason, $(\Delta,{\bf r})$-parking functions are simply called $\Delta$-parking functions. It is shown that the number of $\Delta$-parking functions is less than or equal to the determinant of $\Delta$. Moreover, the definition of $(\Delta,{\bf r})$-recurrent configurations are given for any ${\bf r}\in\mathscr{R}(\Delta)$. It is proved that the set of $(\Delta,{\bf r})$-recurrent configurations is independent of ${\bf r}$ for any ${\bf r}\in\mathscr{R}(\Delta)$. Hence, $(\Delta,{\bf r})$-recurrent configurations are simply called $\Delta$-recurrent configurations. It is obtained that the number of $\Delta$-recurrent configurations is larger than or equal to the determinant of $\Delta$. A simple bijection from $\Delta$-parking functions to $\Delta$-recurrent configurations is established. It follows from this bijection that the number of $\Delta$-parking functions and the number of $\Delta$-recurrent configurations are both equal to the determinant of $\Delta$