×

Conjecture of Wilf: a survey. (English) Zbl 07237998

Barucci, Valentina (ed.) et al., Numerical semigroups. Proceedings of the INdAM international meeting on numerical semigroups, IMNS 2018, Cortona, Italy, September 3–7, 2018. Cham: Springer. Springer INdAM Ser. 40, 39-62 (2020).
Summary: This paper intends to survey the vast literature devoted to a problem posed by Wilf in 1978 which, despite the attention it attracted, remains unsolved. As it frequently happens with combinatorial problems, many researchers who got involved in the search for a solution thought at some point that a solution would be just around the corner, but in the present case that corner has never been reached.
By writing this paper I intend to give the reader a broad approach on the problem and, when possible, connections between the various available results. With the hope of gathering some more information than just using set inclusion, at the end of the paper a slightly different way of comparing results is developed.
For the entire collection see [Zbl 1446.20006].

MSC:

20M14 Commutative semigroups
11D07 The Frobenius problem
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Backelin, J.: On the number of semigroups of natural numbers. Math. Scand. 66(2), 197-215 (1990). https://doi.org/10.7146/math.scand.a-12304 · Zbl 0741.11006 · doi:10.7146/math.scand.a-12304
[2] Barucci, V.: On propinquity of numerical semigroups and one-dimensional local Cohen Macaulay rings. In: Commutative Algebra and Its Applications, pp. 49-60. Walter de Gruyter, Berlin (2009) · Zbl 1177.13005
[3] Barucci, V., Fröberg, R.: One-dimensional almost Gorenstein rings. J. Algebra 188(2), 418-442 (1997). https://doi.org/10.1006/jabr.1996.6837 · Zbl 0874.13018 · doi:10.1006/jabr.1996.6837
[4] Barucci, V., Strazzanti, F.: Dilatations of numerical semigroups. Semigroup Forum 98(2), 251-260 (2019). https://doi.org/10.1007/s00233-018-9922-9 · Zbl 1467.20066 · doi:10.1007/s00233-018-9922-9
[5] Bras-Amorós, M.: Fibonacci-like behavior of the number of numerical semigroups of a given genus. Semigroup Forum 76(2), 379-384 (2008). https://doi.org/10.1007/s00233-007-9014-8 · Zbl 1142.20039 · doi:10.1007/s00233-007-9014-8
[6] Brauer, A.: On a problem of partitions. Am. J. Math. 64, 299-312 (1942). https://doi.org/10.2307/2371684 · Zbl 0061.06801 · doi:10.2307/2371684
[7] Bruns, W., Garcia-Sanchez, P., O’Neill, C., Wilburne, D.: Wilf’s conjecture in fixed multiplicity (2019). arXiv:1903.04342 · Zbl 1481.20207
[8] Curtis, F.: On formulas for the Frobenius number of a numerical semigroup. Math. Scand. 67(2), 190-192 (1990). https://doi.org/10.7146/math.scand.a-12330 · Zbl 0734.11009 · doi:10.7146/math.scand.a-12330
[9] Delgado, M.: IntPic—a GAP package for drawing integers (2017). Version 0.2.3. http://www.gap-system.org/Packages/intpic.html.
[10] Delgado, M.: On a question of Eliahou and a conjecture of Wilf. Math. Z. 288(1-2), 595-627 (2018). https://doi.org/10.1007/s00209-017-1902-3 · Zbl 1486.20076 · doi:10.1007/s00209-017-1902-3
[11] Delgado, M., García-Sánchez, P.A., Morais, J.: Numericalsgps—a GAP package on numerical semigroups (2018). Version number 1.1.7. http://www.gap-system.org/Packages/numericalsgps.html
[12] Dhayni, M.: Problems in numerical semigroups. Ph.D. Thesis, Université d’Angers, 2017. https://www.theses.fr/2017ANGE0041.pdf. · Zbl 1394.20037
[13] Dhayni, M.: Wilf’s conjecture for numerical semigroups. Palest. J. Math. 7(2), 385-396 (2018) · Zbl 1394.20037
[14] Dobbs, D.E., Matthews, G.L.: On a question of Wilf concerning numerical semigroups. In: Focus on Commutative Rings Research, pp. 193-202. Nova Science Publishers, New York (2006) · Zbl 1165.13300
[15] Eliahou, S.: A Graph-Theoretic Approach to Wilf’s Conjecture. Slides presented at the Meeting of the Catalan, Spanish, Swedish Math Societies, 2017. http://www.ugr.es/ semigrupos/Umea-2017/
[16] Eliahou, S.: Wilf’s conjecture and Macaulay’s theorem. J. Eur. Math. Soc. 20(9), 2105-2129 (2018). https://doi.org/10.4171/JEMS/807 · Zbl 1436.20114 · doi:10.4171/JEMS/807
[17] Eliahou, S., Fromentin, J.: Near-misses in Wilf’s conjecture. Semigroup Forum 98(2), 285-298 (2019). https://doi.org/10.1007/s00233-018-9926-5 · Zbl 1467.20070 · doi:10.1007/s00233-018-9926-5
[18] Eliahou, S., Marín-Aragón, D.: Personal communication by Shalom Eliahou (2019)
[19] Fröberg, R., Gottlieb, C., Häggkvist, R.: On numerical semigroups. Semigroup Forum 35(1), 63-83 (1987). https://doi.org/10.1007/BF02573091 · Zbl 0614.10046 · doi:10.1007/BF02573091
[20] Fromentin, J., and Hivert, F.: Exploring the tree of numerical semigroups. Math. Comput. 85(301), 2553-2568 (2016). https://doi.org/10.1090/mcom/3075 · Zbl 1344.20075 · doi:10.1090/mcom/3075
[21] The GAP Group: GAP—groups, algorithms, and programming, version 4.9.1 (2018). https://www.gap-system.org
[22] Geroldinger, A., Halter-Koch, F: Non-unique factorizations: a survey. In: Multiplicative Ideal Theory in Commutative Algebra, pp. 207-226. Springer, New York (2006). https://doi.org/10.1007/978-0-387-36717-0_13 · Zbl 1117.13004 · doi:10.1007/978-0-387-36717-0_13
[23] Kaplan, N.: Counting numerical semigroups by genus and some cases of a question of Wilf. J. Pure Appl. Algebra 216(5), 1016-1032 (2012). https://doi.org/10.1016/j.jpaa.2011.10.038 · Zbl 1255.20054 · doi:10.1016/j.jpaa.2011.10.038
[24] Kaplan, N.: Counting numerical semigroups. Am. Math. Mon. 124(9), 862-875 (2017). https://doi.org/10.4169/amer.math.monthly.124.9.862 · Zbl 1391.20033 · doi:10.4169/amer.math.monthly.124.9.862
[25] Kunz, E.: On the type of certain numerical semigroups and a question of Wilf. Semigroup Forum 93(1), 205-210 (2016). https://doi.org/10.1007/s00233-015-9755-8 · Zbl 1347.20069 · doi:10.1007/s00233-015-9755-8
[26] Kunz, E., Waldi, R.: On the deviation and the type of certain local Cohen-Macaulay rings and numerical semigroups. J. Algebra 478, 397-409 (2017). https://doi.org/10.1016/j.jalgebra.2017.01.041 · Zbl 1393.13044 · doi:10.1016/j.jalgebra.2017.01.041
[27] Matthews, G.L.: On integers nonrepresentable by a generalized arithmetic progression. Integers 5(2), A12 (2005) · Zbl 1139.11301
[28] Moscariello, A., Sammartano, A.: On a conjecture by Wilf about the Frobenius number. Math. Z. 280(1-2), 47-53 (2015). https://doi.org/10.1007/s00209-015-1412-0 · Zbl 1369.11023 · doi:10.1007/s00209-015-1412-0
[29] O’Neill, C., Pelayo, R.: Apéry sets of shifted numerical monoids. Adv. Appl. Math. 97, 27-35 (2018). https://doi.org/10.1016/j.aam.2018.01.005 · Zbl 1402.20066 · doi:10.1016/j.aam.2018.01.005
[30] Ramírez-Alfonsín, J.L.: The Diophantine Frobenius Problem. Oxford Lecture Series in Mathematics and Its Applications, vol. 30. Oxford University Press, Oxford. (2005). https://doi.org/10.1093/acprof:oso/9780198568209.001.0001 · Zbl 1134.11012 · doi:10.1093/acprof:oso/9780198568209.001.0001
[31] Rosales, J.C., García Sánchez, P.A.: Numerical Semigroups. Developments in Mathematics, vol. 20. Springer, New York (2009). https://doi.org/10.1007/978-1-4419-0160-6 · Zbl 1220.20047 · doi:10.1007/978-1-4419-0160-6
[32] Sammartano, A.: Numerical semigroups with large embedding dimension satisfy Wilf’s conjecture. Semigroup Forum 85(3), 439-447 (2012). https://doi.org/10.1007/s00233-011-9370-2 · Zbl 1263.20058 · doi:10.1007/s00233-011-9370-2
[33] Spirito, D.: Wilf’s conjecture for numerical semigroups with large second generator (2017). arXiv:1710.09245
[34] Sylvester, J.J.: Mathematical questions with their solutions. Educ. Times 41, 21 (1884). Solution by W.J. Curran Sharp
[35] Tao, T., Vu, V.: Additive Combinatorics. Cambridge Studies in Advanced Mathematics, vol. 105. Cambridge University Press, Cambridge (2006). https://doi.org/10.1017/CBO9780511755149 · Zbl 1127.11002 · doi:10.1017/CBO9780511755149
[36] Wilf, H.S.: A circle-of-lights algorithm for the “money-changing problem”. Am. Math. Mon. 85(7), 562-565 (1978). https://doi.org/10.2307/2320864 · Zbl 0387.10009 · doi:10.2307/2320864
[37] Zhai, A.: Fibonacci-like growth of numerical semigroups of a given genus. Semigroup Forum 86(3), 634-662 (2013). https://doi.org/10.1007/s00233-012-9456-5 · Zbl 1276.20066 · doi:10.1007/s00233-012-9456-5
[38] Zhao, Y. · Zbl 1204.20080 · doi:10.1007/s00233-009-9190-9
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.