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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
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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
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