  We propose a system of nonlinear integral equations (NLIE) which gives the
free energy of the $U_{q}(widehat{sl}(r+1|s+1))$ Perk-Schultz model. In
contrast with traditional thermodynamic Bethe ansatz equations, our NLIE
contain only r+s+1 unknown functions. In deriving the NLIE, the quantum
(supersymmetric) Jacobi-Trudi and Giambelli formula and a duality for an
auxiliary function play important roles. By using our NLIE, we also calculate
the high temperature expansion of the free energy. General formulae of the
coefficients with respect to arbitrarily rank r+s+1, chemical potentials
$\{\mu_{a}\}$ and q have been written down in terms of characters up to the
order of 5. In particular for specific values of the parameters, we have
calculated the high temperature expansion of the specific heat up to the order
of 40.
