  We introduce the {\it growth partition function} $Z_{\Gamma,G}(t)$ associate
with any cancellative infinite monoid $\Gamma$ with a finite generator system
$G$. It is a power series in $t$ whose coefficients lie in integral Lie-like
space $\mathcal{L}_{\Z}(\Gamma,G)$ in the configuration algebra associated with
the Cayley graph $(\Gamma,G)$. We determine them for homogeneous monoids
admitting left greatest common divisor and right common multiple. Then, for
braid monoids and Artin monoids of finite type, using that formula, we
explicitly determine their limit partition functions $\omega_{\Gamma,G}$.
