  We formulate a new method of performing high-temperature series expansions
for the spin-half Heisenberg model or, more generally, for SU($n$) Heisenberg
model with arbitrary $n$. The new method is a novel extension of the
well-established finite cluster method. Our method emphasizes hidden
combinatorial aspects of the high-temperature series expansion, and solves the
long-standing problem of how to efficiently calculate correlation functions of
operators acting at widely separated sites. Series coefficients are expressed
in terms of cumulants, which are shown to have the property that all deviations
from the lowest-order nonzero cumulant can be expressed in terms of a
particular kind of moment expansion. These ``quasi-moments'' can be written in
terms of corresponding ``quasi-cumulants'', which enable us to calculate
higher-order terms in the high-temperature series expansion. We also present a
new technique for obtaining the low-order contributions to specific heat from
finite clusters.
