  We present three explicit formulas for the number of electronic
configurations in an atom, i.e. the number of ways to distribute $Q$ electrons
in $N$ subshells of respective degeneracies $g_1$, $g_2$, ..., $g_N$. The new
expressions are obtained using the generating-function formalism. The first one
contains sums involving multinomial coefficients. The second one relies on the
idea of gathering subshells having the same degeneracy. A third one also
collects subshells with the same degeneracy and leads to the definition of a
two-variable generating function, allowing the derivation of recursion
relations. Concerning the distribution of population on $N$ distinct subshells
of a given degeneracy $g$, analytical expressions for the first moments of this
distribution are given. The general case of subshells with any degeneracy is
analyzed through the computation of cumulants. A fairly simple expression for
the cumulants at any order is provided, as well as the cumulant generating
function. Using Gram-Charlier expansion, simple approximations of the analyzed
distribution in terms of a normal distribution multiplied by a sum of Hermite
polynomials are given. The Edgeworth expansion has also been tested. Its
accuracy is equivalent to the Gram-Charlier accuracy when few terms are kept,
but it is much more rapidly divergent when the truncation order increases.
While this analysis is illustrated by examples in atomic supershells it also
applies to more general combinatorial problems such as fermion distributions.
