  Suppose that $m$ drivers each choose a preferred parking space in a linear
car park with $n$ spots. In order, each driver goes to their chosen spot and
parks there if possible, and otherwise takes the next available spot if it
exists. If all drivers park successfully, the sequence of choices is called a
parking function. Classical parking functions correspond to the case $m=n$.
  We investigate various probabilistic properties of a uniform parking
function. Through a combinatorial construction termed a parking function
multi-shuffle, we give a formula for the law of multiple coordinates in the
generic situation $m \lesssim n$. We further deduce all possible covariances,
between two coordinates, between a coordinate and an unattempted spot, and
between two unattempted spots. This asymptotic scenario in the generic
situation $m \lesssim n$ is in sharp contrast with that of the special
situation $m=n$.
  A generalization of parking functions called interval parking functions is
also studied, in which each driver is willing to park only in a fixed interval
of spots. We construct a family of bijections between interval parking
functions with $n$ cars and $n$ spots and edge-labeled spanning trees with
$n+1$ vertices and a specified root.

Combinatorics;Probability