Magnetic-substate subtraction ($n_J=d_J-d_{J+1}$)

Let $d_M$ count states of fixed angular momentum projection $M$ in a finite rotational representation, and $n_J$ its spin-$J$ multiplicity. Every spin-$J$ irreducible representation contributes one state at each $M=-J,\dots ,J$. Thus $d_M={\sum }_{J\ge M}{n}_J$ for nonnegative $M$, and subtraction gives $n_J=d_J-d_{J+1}$. Counting occupation states first incorporates Bose-Einstein statistics before decomposing identical-boson angular states.