  This paper is a revision of the combinatorics of fractional exclusion
statistics (FES). More specifically, the following exact statement of the
generalized Pauli principle is derived: for an $N$-particles system exhibiting
FES of extended parameter $g=q/r$ ($q$ and $r$ are co-prime integers such that
$0 < q \leq r$), we found that the allowed occupation number of a state is
smaller than or equal to $r-q+1$ and \emph{not} to $1/g$ whenever $q\neq 1$
and, moreover, the global occupancy shape (merely represented by a partition of
$N$) is admissible if the number of states occupied by at least two particles
is less than or equal to $(N-1)/r$ ($N \equiv 1 \pmod r$). These counting rules
allow distinguishing infinitely many families of FES systems depending on the
parameter $g$ and the size $N$. As an application of the main result, we study
the probability distributions of occupancy configurations. For instance, the
number of occupied states is found to be a hypergeometric random variable.
Closed-form expressions for the expectation values and variances in the
thermodynamic limit are presented. By way of comparison, we obtain parallel
results regarding the Gentile intermediate statistics and demonstrate subtle
similarities and contrasts with respect to FES.
