We introduce the growth partition function ZΓ,G(t) associate with any cancellative infinite monoid Γ with a finite generator system G. It is a power series in t whose coefficients lie in integral Lie-like space
Nearby in the stack
LZ(Γ,G)
in the configuration algebra associated with the Cayley graph
(Γ,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