  Generalized Word Length Pattern (GWLP) is an important and widely-used tool
for comparing fractional factorial designs. We consider qualitative factors,
and we code their levels using the roots of the unity. We write the GWLP of a
fraction ${\mathcal F}$ using the polynomial indicator function, whose
coefficients encode many properties of the fraction. We show that the
coefficient of a simple or interaction term can be written using the counts of
its levels. This apparently simple remark leads to major consequence, including
a convolution formula for the counts. We also show that the mean aberration of
a term over the permutation of its levels provides a connection with the
variance of the level counts. Moreover, using mean aberrations for symmetric
$s^m$ designs with $s$ prime, we derive a new formula for computing the GWLP of
${\mathcal F}$. It is computationally easy, does not use complex numbers and
also provides a clear way to interpret the GWLP. As case studies, we consider
non-isomorphic orthogonal arrays that have the same GWLP. The different
distributions of the mean aberrations suggest that they could be used as a
further tool to discriminate between fractions.
