  Permutation Entropy (PE) is a powerful tool for quantifying the complexity of
a signal which includes measuring the regularity of a time series.
Additionally, outside of entropy and information theory, permutations have
recently been leveraged as a graph representation, which opens the door for
graph theory tools and analysis. Despite the successful application of
permutations in a variety of scientific domains, permutations require a
judicious choice of the delay parameter $\tau$ and dimension $n$. However, $n$
is typically selected between $4$ and $8$ with $5$ or $6$ giving optimal
results for the majority of systems. Therefore, in this work, we focus on
choosing the delay parameter while giving some general guidance on the
appropriate selection of $n$ based on a statistical analysis of the permutation
distribution. Selecting $\tau$ is often accomplished using trial and error
guided by the expertise of domain scientists. However, in this paper, we show
how persistent homology, a commonly used tool from Topological Data Analysis
(TDA), provides methods for the automatic selection of $\tau$. We evaluate the
successful identification of a suitable $\tau$ from our TDA-based approach by
comparing our results to both expert-suggested parameters from published
literature and optimized parameters (if possible) for a wide variety of
dynamical systems.
