On vector parking functions and q-analogue

Abstract: In 2000, it was demonstrated that the set of $x$-parking functions of length $n$, where $x$=($a,b,...,b$) $\in \mathbbm{N}^n$, is equivalent to the set of rooted multicolored forests on [$n$]={1,...,$n$}. In 2020, Yue Cai and Catherine H. Yan systematically investigated the properties of rational parking functions. Subsequently, a series of Context-free grammars possessing the requisite property were introduced by William Y.C. Chen and Harold R.L. Yang in 2021. %An Abelian-type identity is derived from a comparable methodology and grammatical framework. %Leveraging a comparable methodology and grammatical framework, an Abelian-type identity is derived herein. In this paper, I discuss generalized parking functions in terms of grammars. The primary result is to obtain the q-analogue about the number of '1's in certain vector parking functions with the assistance of grammars.