Korepin–Izergin–Baxter ferromagnetic-string probability formula

For the even-site ground state $|\Psi^{(2n)}\rangle$, define

$$\langle O\rangle_{2n}=\frac{\langle\Psi^{(2n)}|O|\Psi^{(2n)}\rangle}{\langle\Psi^{(2n)}|\Psi^{(2n)}\rangle},$$

and set $\alpha_i=(1+\sigma_i^z)/2$. Korepin–Izergin–Baxter ferromagnetic-string formula. The probabilities of formation of ferromagnetic strings satisfy

$$\frac{\langle\alpha_1\alpha_2\cdots\alpha_{k-1}\rangle_{2n}}{\langle\alpha_1\alpha_2\cdots\alpha_k\rangle_{2n}} =\frac{(2k-2)!(2k-1)!(2n+k-1)!(n-k)!}{(k-1)!(3k-2)!(2n-k)!(n+k-1)!}.$$

The source calls this a simple formula and attributes it to KIB; its status is not established by the supplied text.