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.