  One-dimensional repulsive delta-function bose system is studied. By only
using the Bethe ansatz equation, n-particle partition functions are exactly
calculated. From this expression for the n-particle partition function, the
n-particle cluster integral is derived. The results completely agree with those
of the thermal Bethe ansatz (TBA). This directly proves the validity of the
TBA. The theory of partitions and graphs is used to simplify the discussion.
