A general method is proposed to obtain the distribution of the total quantum number $M$ for a set of $N$ identical fermions with momentum $j$, which is a cornerstone of the nuclear shell model. This can be performed using a recursive procedure on $N$, yielding closed-form expressions, which are found to be linear combinations of piecewise polynomials. We also highlight and implement in that framework two three-term recurrence relations over $N$, more convenient than Talmi's five-term recurrence which has nevertheless already proved its worth in the past. In addition, the current approach allows one to consider both integer and half-integer values of $j$ on the same footing. The technique is illustrated by detailed examples, corresponding to $N=3$ to 6 fermions.

• Received 26 September 2023
• Revised 18 December 2023
• Accepted 10 January 2024

DOI:https://doi.org/10.1103/PhysRevC.109.024306