  The many $2^k$-pole boson states, $|N_kv_k\alpha_k I_kM_k>$ with $k=2,3$,
realize the irreducible representation (IR) for the group reduction chains
$SU(2k+1)\supset R_{2k+1}\supset R_3\supset R_2$. They have been analytically
studied and widely used for the description of nuclear systems. However, no
analytical expression for the degeneracy $d_v(I)$ of the $R_{2k+1}$'s IR,
determined by the reduction $R_{2k+1}\supset R_3$, is available. Thus, the
number of distinct values taken by $\alpha_k$ has been so far obtained by
solving some complex equations. Here we derive analytical expressions for the
degeneracy $d_v(I)$ characterizing the octupole and quadrupole boson states,
respectively. The merit of this work consists of the fact that it completes the
analytical expressions for the $2^k$-pole boson basis.
