  We exploit the grassmannian nature of the variables involved in the path
integral expression of the grand canonical partition function for
self--interacting fermionic models to show, in one-space dimension, a general
relation among the terms of it expansion in the high temperature limit and a
combination of co-factors of a suitable matrix with commuting entries. As an
application, we apply this framework to calculate the exact coefficients, up to
order \beta^3, of the expansion of the grand canonical partition function for
the Hubbard model in d=(1+1) in the high temperature limit. The results are
valid for any set of parameters that characterize the model.
