On the enumeration of generalized parking functions (Q2716644)

A generalized parking function is defined as follows: let \(\vec {x}=(x_1,\ldots,x_n) \in \mathbb{N} ^n\) be a vector of positive integers. Then an \(\vec x\)-parking function is a sequence \((a_1,\ldots,a_n)\) of positive integers whose non-decreasing rearrengment \(b_1 \leq \ldots \leq b_n\) satisfies \(b_i \leq x_1 + \cdots + x_i.\) The case \(x_i=1\) for all \(i\) supplies the notion of parking function introduced in 1966 by Konheim and Weiss in connection of occupancy problems in computer science. The number of generalized parking functions got quite an attention recently, but it is far from a complete solution. This paper solves far-reaching special cases by means of combinatorial arguments as well as of the generating function method.