# zbMATH — the first resource for mathematics

Counting numerical semigroups by genus and even gaps. (English) Zbl 06775838
Summary: Let $$n_g$$ be the number of numerical semigroups of genus $$g$$. We present an approach to compute $$n_g$$ by using even gaps, and the question: Is it true that $$n_{g + 1} > n_g$$? is investigated. Let $$N_\gamma(g)$$ be the number of numerical semigroups of genus $$g$$ whose number of even gaps equals $$\gamma$$. We show that $$N_\gamma(g) = N_\gamma(3 \gamma)$$ for $$\gamma \leq \lfloor g/3 \rfloor$$ and $$N_\gamma(g) = 0$$ for $$\gamma > \lfloor 2g/3 \rfloor$$; thus the question above is true provided that $$N_\gamma(g + 1) > N_\gamma(g)$$ for $$\gamma = \lfloor g/3 \rfloor + 1, \ldots, \lfloor 2g/3 \rfloor$$. We also show that $$N_\gamma(3 \gamma)$$ coincides with $$f_\gamma$$, the number introduced by Bras-Amorós (2012) in connection with semigroup-closed sets. Finally, the stronger possibility $$f_\gamma \sim \varphi^{2 \gamma}$$ arises being $$\varphi = (1 + \sqrt{5})/2$$ the golden number.

##### MSC:
 20M14 Commutative semigroups 05A15 Exact enumeration problems, generating functions 05A16 Asymptotic enumeration
Full Text:
##### References:
 [1] Blanco, V.; García-Sánchez, P. A.; Puerto, J., Computing numerical semigroups with short generating functions, Int. J. Algebra Comput., 21, 1217-1235, (2011) · Zbl 1250.20048 [2] Blanco, V.; Rosales, J. C., The set of numerical semigroups of a given genus, Semigroup Forum, 85, 255-267, (2012) · Zbl 1263.20057 [3] Bras-Amorós, M., Fibonacci-like behavior of the number of numerical semigroups of a given genus, Semigroup Forum, 76, 379-384, (2008) · Zbl 1142.20039 [4] Bras-Amorós, M., Bounds on the number of numerical semigroups of a given genus, J. Pure Appl. Algebra, 213, 6, 997-1001, (2009) · Zbl 1169.05300 [5] Bras-Amorós, M., The ordinarization transform of a numerical semigroup and semigroups with a large number of intervals, J. Pure Appl. Algebra, 216, 2507-2518, (2012) · Zbl 1261.20072 [6] Bras-Amorós, M., Semigroups and codes, (Martinez-Moro, E., Algebraic Geometry Modeling in Information Theory, (2013), Word Scientific), 167-218 · Zbl 1352.94090 [7] Bras-Amorós, M.; Bulygin, S., Towards a better understanding of the semigroup tree, Semigroup Forum, 79, 561-574, (2009) · Zbl 1230.05018 [8] Bras-Amorós, M.; de Mier, A., Representation of numerical semigroups by Dyck paths, Semigroup Forum, 75, 676-681, (2007) · Zbl 1128.20046 [9] M. Bras-Amorós, J. Fernández-González, Computation of numerical semigroups by means of seeds, Math. Comput. (in press). [10] D’Anna, M.; Strazzanti, F., The numerical duplication of a numerical semigroup, Semigroup Forum, 87, 149-160, (2013) · Zbl 1282.20065 [11] M. Delgado, P.A. García-Sánchez, J. Morais, NumericalSgps A package for numerical semigroups , Version 1.0.1 2015, http://www.fc.up.pt/cmup/mdelgado/numericalsgps. [12] Eisenbud, D.; Harris, J., Existence, decomposition and limits of certain Weierstrass points, Invent. Math., 87, 495-515, (1987) · Zbl 0606.14014 [13] Elizalde, S., Improved bounds on the number of numerical semigroups of a given genus, J. Pure Appl. Algebra, 214, 1862-1873, (2010) · Zbl 1185.05008 [14] Fromentin, J.; Hivert, F., Exploring the tree of numerical semigroups, Math. Comp., 85, 301, 2553-2568, (2016) · Zbl 1344.20075 [15] Garcia, A., Weights of Weierstrass points in double covering of curves of genus one or two, Manuscripta Math., 55, 419-432, (1986) · Zbl 0603.14014 [16] García-Sánchez, P. A.; Rosales, J. C., (Numerical Semigroups, Developments in Mathematics, vol. 20, (2009), Springer New York) · Zbl 1220.20047 [17] Gu, Z.; Tang, X., The doubles of one half of a numerical semigroup, J. Number Theory, 163, 375-384, (2016) · Zbl 1344.20076 [18] N. Kaplan, Counting numerical semigroups, Amer. Math. Monthly (in press). · Zbl 1391.20033 [19] Kaplan, N., Counting numerical semigroups by genus and some cases of a question of Wilf, J. Pure Appl. Algebra, 216, 1016-1032, (2012) · Zbl 1255.20054 [20] Kato, T., On criteria of $$\widetilde{g}$$-hyperellipticity, Kodai Math. J., 2, 275-285, (1979) · Zbl 0425.30038 [21] Komeda, J., On primitive shubert indices of genus $$g$$ and weight $$g - 1$$, J. Math. Soc. Japan, 43, 3, 437-445, (1991) · Zbl 0753.14028 [22] Komeda, J., On Weierstrass semigroups of double coverings of genus three curves, Semigroup Forum, 83, 479-488, (2011) · Zbl 1244.14025 [23] O’Dorney, E., Degree asymptotics of the numerical semigroup tree, Semigroup Forum, 87, 601-616, (2013) · Zbl 1319.20050 [24] Oliveira, G.; Pimentel, F. L.R., On Weierstrass semigroups of double covering of genus two curves, Semigroup Forum, 77, 152-162, (2008) · Zbl 1161.14023 [25] Oliveira, G.; Torres, F.; Villanueva, J., On the weight of numerical semigroups, J. Pure Appl. Algebra, 214, 1955-1961, (2010) · Zbl 1194.14048 [26] Pellikaan, R.; Torres, F., On Weierstrass semigroups and the redundancy of improved geometric Goppa codes, IEEE Trans. Inform. Theory, 45, 7, 2512-2519, (1999) · Zbl 0960.94027 [27] Ramírez-Alfonsín, J. L., The Diophantine Frobenius Problem, Vol. 30, (2005), Oxford Univ. Press · Zbl 1134.11012 [28] Robles-Pérez, A. M.; Rosales, J. C.; Vasco, P., The doubles of a numerical semigroup, J. Pure Appl. Algebra, 213, 387-396, (2009) · Zbl 1166.20054 [29] Rosales, J. C.; García-Sánchez, P. A.; García-Sánchez, J. I.; Urbano-Blanco, J. M., Proportionally modular Diophantine inequalities, J. Number Theory, 103, 281-294, (2003) · Zbl 1039.20036 [30] Selmer, E. S., On the linear Diophantine problem of Frobenius, J. Reine Angew. Math., 293/294, 1-17, (1977) · Zbl 0349.10009 [31] N.J.A. Sloane, “The On-Line Encyclopedia of Integer Sequences”, A007323, 2009. http://www.research.att.com/ njas/sequences/. · Zbl 1044.11108 [32] Strazzanti, F., Minimal genus of a multiple and Frobenius number of a quotient of a numerical semigroup, Int. J. Algebra Comput., 25, 1043-1053, (2015) · Zbl 1344.20078 [33] Torres, F., Weierstrass points and double coverings of curves. with application: symmetric numerical semigroups which cannot be realized As weierstr ass semigroups, Manuscripta Math., 83, 39-58, (1994) · Zbl 0838.14025 [34] Torres, F., On $$\gamma$$-hyperelliptic numerical semigroups, Semigroup Forum, 55, 364-379, (1997) · Zbl 0931.14017 [35] Zhai, A., Fibonacci-like growth of numerical semigroups of a given genus, Semigroup Forum, 86, 634-662, (2013) · Zbl 1276.20066
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.