Parking functions on toppling matrices

Let $\mathrm{\Delta }$ be an integer $n\times n$-matrix which satisfies the conditions: $\det \mathrm{\Delta }\mathrm{\ne }0$, ${\mathrm{\Delta }}_{ij}\le 0\text{ for }i\mathrm{\ne }j,$ and there exists a vector $\mathbf{r}=(r_1,\dots ,r_n)>0$ such that ${\bf r}\Delta \geq 0$\mathrm{.}Herethenotation${\bf r}> 0$meansthat$r_i>0$forall$i$,and${\bf r}\geq {\bf r}‘$meansthat$r_i\geq r’_i$forevery$i$\mathrm{.}Let$\mathscr{R}(\Delta)$bethesetofvectors${\bf r}$suchthat${\bf r}>0$and${\bf r}\Delta\geq 0$\mathrm{.}Inthispaper,$(\Delta,{\bf r})$-parking functions are defined for any $\mathbf{r}\in \mathcal{R}(\mathrm{\Delta })$. It is proved that the set of $(\mathrm{\Delta },\mathbf{r})$-parking functions is independent of $\mathbf{r}$ for any ${\bf r}\in\mathscr{R}(\Delta)$\mathrm{.}Forthisreason,$(\Delta,{\bf r})$-parking functions are simply called $\mathrm{\Delta }$-parking functions. It is shown that the number of $\mathrm{\Delta }$-parking functions is less than or equal to the determinant of $\mathrm{\Delta }$. Moreover, the definition of $(\mathrm{\Delta },\mathbf{r})$-recurrent configurations are given for any $\mathbf{r}\in \mathcal{R}(\mathrm{\Delta })$. It is proved that the set of $(\mathrm{\Delta },\mathbf{r})$-recurrent configurations is independent of $\mathbf{r}$ for any $\mathbf{r}\in \mathcal{R}(\mathrm{\Delta })$. Hence, $(\Delta,{\bf r})$-recurrentconfigurationsaresimplycalled$\Delta$-recurrent configurations. It is obtained that the number of $\mathrm{\Delta }$-recurrent configurations is larger than or equal to the determinant of $\mathrm{\Delta }$. A simple bijection from $\mathrm{\Delta }$-parking functions to $\mathrm{\Delta }$-recurrent configurations is established. It follows from this bijection that the number of $\mathrm{\Delta }$-parking functions and the number of $\mathrm{\Delta }$-recurrent configurations are both equal to the determinant of $\mathrm{\Delta }$.