Because of the SU(2) algebra, we know that the eigenvalues of $\hat{P}^2$ must be of the form $p(p+1)$, with $p$ either integer or half-integer, and the eigenvalues of $\hat{P}_z$ are $m_p$ with $p \geq | m_p|$, with $m_p$ also integer or half-integer.

But because $\hat{P}_z = (1/2)(\hat{N}-\Omega)$, we know that for $N$ particles the value $m_p = (N-\Omega)/2$. Furthermore, the values of $m_p$ range from $-\Omega/2$ (for $N=0$) to $+\Omega/2$ (for $N=2\Omega$, with all states filled).

We deduce the maximal $p = \Omega/2$ and for a given $n$ the values range of $p$ range from $|N-\Omega|/2$ to $\Omega/2$ in steps of 1 (for an even number of particles)

Following Racah we introduce the notation $p = (\Omega - v)/2$ where $v = 0, 2, 4,..., \Omega - |N-\Omega|$ With this it is easy to deduce that the eigenvalues of the pairing Hamiltonian are $$-G(N-v)(2\Omega +2-N-v)/4$$ This also works for $N$ odd, with $v= 1,3,5, \dots$.