= Magnetic-substate subtraction
{title2=$n_J=d_J-d_{J+1}$}
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 $M=-J,\ldots,J$. Thus $d_M=\sum_{J\geq M}n_J$ 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.
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