On the number of factorizations of an element in an atomic monoid. (Q1866180)

Let \(S\) be a finitely generated commutative cancellative reduced monoid. For \(s \in S\), let \(\eta (s)\) be the number of distinct factorizations of \(s\) into irreducible elements of \(S\). Then there are positive constants \(A(s) \in \mathbb Q\) and \(r(s) \in \mathbb N\) such that \(\eta (s) = A(s) n^{r(s)-1} + O(n^{r(s)-2})\) [\textit{F. Halter-Koch}, Ark. Mat. 31, 297--305 (1993; Zbl 0792.11042)]. The authors provide algorithms for the computation of \(A(s)\) and \(r(s)\) in terms of generators and defining relations of \(S\).