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Parking functions on toppling matrices

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