  This is a paper in the intersection of time series analysis and complexity
theory that presents new results on permutation complexity in general and
permutation entropy in particular. In this context, permutation complexity
refers to the characterization of time series by means of ordinal patterns
(permutations), entropic measures, decay rates of missing ordinal patterns, and
more. Since the inception of this \textquotedblleft ordinal\textquotedblright\
methodology, its practical application to any type of scalar time series and
real-valued processes have proven to be simple and useful. However, the
theoretical aspects have remained limited to noiseless deterministic series and
dynamical systems, the main obstacle being the super-exponential growth of
visible permutations with length when randomness (also in form of observational
noise) is present in the data. To overcome this difficulty, we take a new
approach through complexity classes, which are precisely defined by the growth
of visible permutations with length, regardless of the deterministic or noisy
nature of the data. We consider three major classes: exponential, sub-factorial
and factorial. The next step is to adapt the concept of Z-entropy to each of
those classes, which we call permutation entropy because it coincides with the
conventional permutation entropy on the exponential class. Z-entropies are a
family of group entropies, each of them extensive on a given complexity class.
The result is a unified approach to the ordinal analysis of deterministic and
random processes, from dynamical systems to white noise, with new concepts and
tools. Numerical simulations show that permutation entropy discriminates time
series from all complexity classes.
