Some aspects of (r,k)-parking functions

Abstract

An \emph{$(r,k)$-parking function} of length $n$ may be defined as a sequence $(a_1,\dots ,a_n)$ of positive integers whose increasing rearrangement $b_1\le \cdots \le b_n$ satisfies $b_i\le k+(i-1)r$. The case $r=k=1$ corresponds to ordinary parking functions. We develop numerous properties of $(r,k)$-parking functions. In particular, if $F_n^{(r,k)}$ denotes the Frobenius characteristic of the action of the symmetric group ${\mathfrak{S}}_n$ on the set of all $(r,k)$-parking functions of length $n$, then we find a combinatorial interpretation of the coefficients of the power series ${\left({\sum }_{n\ge 0}{F}_n^{(r,1)}{t}^n\right)}^k$ for any $k\in \mathbb{Z}$. When $k>0$, this power series is just ${\sum }_{n\ge 0}{F}_n^{(r,k)}{t}^n$; when $k<0$, we obtain a dual to $(r,k)$-parking functions. We also give a $q$-analogue of this result. For fixed $r$, we can use the symmetric functions $F_n^{(r,1)}$ to define a multiplicative basis for the ring $\Lambda$ of symmetric functions. We investigate some of the properties of this basis.

References in corpus (4)

• Combinatorics of r-Dyck paths, r-Parking functions, and the r-Tamari lattices
• Interlaced rectangular parking functions
• Rectangular Schroder Parking Functions Combinatorics
• Bounded Height Interlaced Pairs of Parking Functions