  In the past decade, the use of ordinal patterns in the analysis of time
series and dynamical systems has become an important and rich tool. Ordinal
patterns (otherwise known as a permutation patterns) are found in time series
by taking $n$ data points at evenly-spaced time intervals and mapping them to a
length-$n$ permutation determined by relative ordering. The frequency with
which certain patterns occur is a useful statistic for such series; however,
the behavior of the frequency of pattern occurrence is unstudied for most
models. We look at the frequency of pattern occurrence in random walks in
discrete time and, applying combinatorial methods, we characterize those
patterns that have equal frequency, regardless of probability distribution.
