Statistics of multi-electron states and $J$-levels in atomic configurations

The number and nature of atomic configurations are cornerstones of atomic spectroscopy, especially for the calculation of hot-plasma radiative properties. The knowledge of the distributions of magnetic quantum number $M$ and angular momentum $J$ for $N$ identical fermions in a subshell with half-integer spin $j$ is a prerequisite to the determination of the structure of such configurations. The problem is rather complicated, since the possible occurrence of a specific values of $J$ is governed by the Pauli exclusion principle. Several methods, such as generating functions, recurrence relations or algebraic number theory, for instance via Gaussian polynomials, have proven effective in addressing this issue. However, up to now, no general formula was known. In the present work, we present exact and compact explicit formulas for the number of atomic configurations and for the distributions of the total magnetic quantum number $M$ and angular momentum $J$.