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The frequency of pattern occurrence in random walks

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.