  A quite general finite-size chain of fermions with $N$ internal degrees of
freedom (flavors) and $O(N)$ symmetry is considered. In the case of the free
boundary condition, we prove that the ground state in the invariant sector
having exactly $m$ flavors with an odd particle number is represented by a
single rank-$m$ antisymmetric multiplet. For the even-length chains, its
particle-hole quantum number (if it's a good one) is given by the parity of the
$m$. For the odd-length chains, the particle-hole symmetry leads to the twofold
degeneracy among the conjugate multiplets. Similar statements are proven for
the $O(N)$ mixed-spin chains in antisymmetric representations. The results are
extended to the long-range interacting fermions and (partially) to the
translation invariant chains.
