On maximal parsings of strings

Given a sequence, we consider the maximum number of distinct phrases in any parsing; this definition of complexity is invariant under string reversal. We show that the Lempel-Ziv (1976, 1978) parsings can vary under reversal by a factor on the order of the log of the sequence length. We give two interpretations of maximal parsing, show that they are not equivalent and that one lacks a plausible monotonicity property.