On vector parking functions and q-analogue

TL;DR

This paper introduces a q-analogue for counting specific vector parking functions, utilizing grammatical methods to extend previous combinatorial results and explore properties of generalized parking functions.

Contribution

It develops a new q-analogue framework for vector parking functions using grammatical techniques, advancing the combinatorial understanding of these functions.

Findings

Derived a q-analogue for the number of '1's in vector parking functions

Extended grammatical methods to generalized parking functions

Connected parking functions with rooted multicolored forests

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.