The maximal denumerant of a numerical semigroup

Given a numerical semigroup S=aOE (c) a (0),a (1),a (2),aEuro broken vertical bar,a (t) > and naS, we consider the factorization n=c (0) a (0)+c (1) a (1)+a <-+c (t) a (t) where c (i) a parts per thousand yen0. Such a factorization is maximal if ac (i) is a maximum over all such factorizations of n. We provide an algorithm for computing the maximum number of maximal factorizations possible for an element in S, which is called the maximal denumerant of S. We also consider various cases that have connections to the Cohen-Macualay and Gorenstein properties of associated graded rings for which this algorithm simplifies.