Denumerants of 3-numerical semigroups Aguil´o-GostF. 1⋆ , Garc´ ıa-S´ anchez P. 2⋆⋆ , and Llena D. 3⋆⋆⋆ 1 Departament MA–IV. Universitat Polit` ecnica de Catalunya. matfag@ma4.upc.edu 2 Departamento de ´ Algebra. Universidad de Granada. pedro@ugr.es 3 Departamento de Matem´ aticas. Universidad de Almer´ ıa. dllena@ual.es Abstract. Denumerants of numerical semigroups are known to be difficult to ob- tain, even with small embedding dimension of the semigroups. In this work we give some results on denumerants of 3-semigroups S = 〈a, b, c〉. The time efficiency of the resulting algorithms range from O(1) to O(c). Closed expressions are obtained under certain conditions. Key words: Denumerant, L-shapes, numerical semigroup, factorization. 1 Introduction Let N 0 be the set of non negative integers. Given a set A = {a 1 ,...,a n }⊂ N 0 , gcd(a 1 ,...,a n ) = 1, the n-numerical semigroup S = S (A) generated by A is defined by 〈a 1 ,...,a m 〉 = {x 1 a 1 + ··· + x n a n : x 1 ,...,x n ∈ N 0 }. If A is a minimal set of generators then S has embedding dimension e(S )= n. An element m ∈ S has a factorization (t 1 ,...,t n ) in S if m = t 1 a 1 + ··· + t n a n . The set of factorizations of m in S is denoted by F (m, S )= {(x 1 ,...,x n ) ∈ N n 0 : x 1 a 1 + ··· + x n a n = m}. The denumerant of m in S is the cardinality d(m, S )= |F (m, S )|. In this work we give some results on denumerants of generic elements of embedding dimension three numerical semigroups S = 〈a, b, c〉. Algorithms of time–cost ranging from O(1) to O(c) (in the worst case) are also derived. We use minimum distance diagrams related to S as a main tool. In particular, we use the main results given in [1]. ⋆ Work supported by the CICYT and the ‘European Regional Development Fund’ under project MTM2011-28800-C02-01 and Ag` encia de Gesti´ o d’Ajuts Universitaris i de Recerca under project 2009SGR1387. ⋆⋆ Investigaci´ on financiada por MTM2010-15595, FQM-343, FQM-5849 y fondos FEDER ⋆⋆⋆ Investigaci´ on financiada por MTMT2010-15595, FQM-343 y fondos FEDER