  Based on the exact solution of one-dimensional Fermi gas systems with $SU(n)$
symmetry in a hard wall, we demonstrate that we are able to sort the ordering
of the lowest energy eigenvalues of states with all allowed permutation
symmetries, which can be solely marked by certain quantum numbers in the Bethe
ansatz equations. Our results give examples beyond the scope of the generalized
Lieb-Mattis theorem, which can only compare the ordering of energy levels of
states belonging to different symmetry classes if they are comparable according
to the pouring principle. In the strongly interacting regime, we show that the
ordering of energy levels can be determined by an effective spin-exchange model
and extend our results to the non-uniform system trapped in the harmonic
potential.
