  The finite XXZ Heisenberg spin chain with twisted boundary conditions was
considered. For the case of even number of sites $N$, anisotropy parameter -1/2
and twisting angle $2 \pi /3$ the Hamiltonian of the system possesses an
eigenvalue $-3N/2$. The explicit form of the corresponding eigenvector was
found for $N \le 12$. Conjecturing that this vector is the ground state of the
system we made and verified several conjectures related to the norm of the
ground state vector, its component with maximal absolute value and some
correlation functions, which have combinatorial nature. In particular, the
squared norm of the ground state vector is probably coincides with the number
of half-turn symmetric alternating sign $N \times N$ matrices.
