The Bethe Ansatz

We put \(M_1^< = N/2 -2\), \(M_1^\infty = 1\). The equations for the limiting quantum number fall back onto the \(M \rightarrow M-1\) ones. The limiting quantum numbers are \(I^{1,\infty} = \frac{N}{4} + \frac{3}{2}\) so \(I^{\mbox{max}} = \frac{N}{4} + \frac{1}{2}\), so we have to put \(N/2 - 2\) quantum numbers in \(N/2 + 2\) slots, yielding \(\frac{(N+4) (N+2) N (N-2)}{384}\) states, which are the four-spinon states in this sector.