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We present three explicit formulas for the number of electronicconfigurations in an atom, i.e. the number of ways to distribute $Q$ electronsin $N$ subshells of respective degeneracies $g_1$, $g_2$, ..., $g_N$. The newexpressions are obtained using the generating-function formalism. The first onecontains sums involving multinomial coefficients. The second one relies on theidea of gathering subshells having the same degeneracy. A third one alsocollects subshells with the same degeneracy and leads to the definition of atwo-variable generating function, allowing the derivation of recursionrelations. Concerning the distribution of population on $N$ distinct subshellsof a given degeneracy $g$, analytical expressions for the first moments of thisdistribution are given. The general case of subshells with any degeneracy isanalyzed through the computation of cumulants. A fairly simple expression forthe cumulants at any order is provided, as well as the cumulant generatingfunction. Using Gram-Charlier expansion, simple approximations of the analyzeddistribution in terms of a normal distribution multiplied by a sum of Hermitepolynomials are given. The Edgeworth expansion has also been tested. Itsaccuracy 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 alsoapplies to more general combinatorial problems such as fermion distributions.

Analytical and numerical expressions for the number of atomic configurations contained in a supershell