  We examine the aggregate behavior of one-dimensional random walks in a model
known as (one-dimensional) Internal Diffusion Limited Aggregation. In this
model, a sequence of $n$ particles perform random walks on the integers,
beginning at the origin. Each particle walks until it reaches an unoccupied
site, at which point it occupies that site and the next particle begins its
walk. After all walks are complete, the set of occupied sites is an interval of
length $n$ containing the origin. We show the probability that $k$ of the
occupied sites are positive is given by an Eulerian probability distribution.
Having made this connection, we use generating function techniques to compute
the expected run time of the model.
