%I M1064
%S 1,1,2,4,7,12,23,39,67,118,204,343,592,1001,1693,2857,4806,8045,13467,
%T 22464,37396,62194,103246,170963,282828,467224,770832,1270267,2091030,
%U 3437839,5646773,9266788,15195070,24896206,40761087,66687201,109032500,178158289
%N The number of numerical semigroups of "genus" n; conjecturally also the number of power sum bases for symmetric functions in n variables.
%C From Don Zagier's email of Apr 11 1994: (Start)
%C Given n, one knows that the field of symmetric functions in n variables a_1,...,a_n is the field Q(sigma_1,...,sigma_n), where sigma_i is the ith elementary symmetric polynomial. Here one has no choice, because sigma_i=0 for i>n and fewer than n sigma's would not suffice.
%C But, by Newton's formulas, the field is also given as Q(s_1,...,s_n) where s_i is the ith power sum, and now one can ask whether some other sequence s_{j_1},...,s_{j_n} (0<j_1<...<j_n) also works.
%C For n=1 the only possibility is clearly s_1, since Q(s_i) = Q(a^i) does not coincide with Q(a) for i>1, but for n=2 one has two possibilities Q(s_1,s_2) or Q(s_1,s_3), since from s_1=a+b and s_3=a^3+b^3 one can reconstruct s_2 = (s_1^3+2s_3)/3s_1.
%C Similarly, for n=3 one has the possibilities (123), (124), (125), and (135) (the formula in the last case is s_2 = (s_1^5+5s_1^2s_36s_5)/5(s_1^3s_3); one can find the corresponding formulas in the other cases easily) and for n=4 there are 7: 1234, 1235, 1236, 1237, 1245, 1247, and 1357.
%C A theorem of Kakutani (I do not know a reference) says that the sequences which occur are exactly the finite subsets of N whose complements are additive semigroups (for instance, the complement of {1,2,4,7} is 3,5,6,8,9,..., which is closed under addition).
%C This is a really beautiful theorem. I wrote a simple program to count the sets of cardinality n which have the property in question for n = 1, ..., 16. (End)
%C This sequence relates to numerical semigroups, which are basic fundamental objects but little known: A numerical semigroup S < N is defined by being: closed under addition, contains zero and N \ S is finite. [John McKay, Jun 09 2011]
%C The theorem alluded to in the email by Zagier is due to Kakeya, not Kakutani (see references.) The theorem states that if a sequence of n positive integers k1, k2,..., kn forms the complement of a numerical semigroup, then the power sums p_k1, p_k2,..., p_kn forms a basis for the rational function field of symmetric functions in n variables. Kakeya conjectures that every power sum basis of the symmetric functions has this property, but this is still an open problem. Thanks to user Gjergji Zaimi on Math Overflow for the references. [Trevor Hyde, Oct 18 2018]
%D Sean Clark, Anton Preslicka, Josh Schwartz and Radoslav Zlatev, Some combinatorial conjectures on a family of toric ideals: A report from the MSRI 2011 Commutative Algebra graduate workshop.
%D S. Kakeya, On fundamental systems of symmetric functions, Jap. J. Math., 2, (1925), 6980.
%D S. Kakeya, On fundamental systems of symmetric functions II, Jap. J. Math., 4, (1927), 7785.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Maria BrasAmoros and J. Fromentin, <a href="/A007323/b007323.txt">Table of n, a(n) for n = 0..60</a>
%H Matheus Bernardini, and Fernando Torres, <a href="https://arxiv.org/abs/1612.01212">Counting numerical semigroups by genus and even gaps</a>, arXiv:1612.01212 [math.CO], 20162017.
%H Matheus Bernardini, and Fernando Torres, <a href="https://doi.org/10.1016/j.disc.2017.08.001">Counting numerical semigroups by genus and even gaps</a>, Discrete Mathematics 340.12 (2017): 28532863.
%H Victor Blanco, Justo Puerto, <a href="http://arxiv.org/abs/0901.1228">Computing the number of numerical semigroups using generating functions</a>, arXiv:0901.1228 [math.CO], 2009.
%H Maria BrasAmoros, <a href="http://crisesdeim.urv.cat/~mbras/">Home Page</a> [Has many of these references]
%H M. BrasAmoros, <a href="http://dx.doi.org/10.1007/s0023300790148">FibonacciLike Behavior of the Number of Numerical Semigroups of a Given Genus</a>, Semigroup Forum, 76 (2008), 379384. See <a href="https://arxiv.org/abs/1706.05230">also</a>, arXiv:1706.05230 [math.NT], 2017.
%H M. BrasAmoros, <a href="http://dx.doi.org/10.1016/j.jpaa.2008.11.012">Bounds on the Number of Numerical Semigroups of a Given Genus</a>, Journal of Pure and Applied Algebra, vol. 213, n. 6 (2009), pp. 9971001. arXiv:0802.2175.
%H M. BrasAmoros and S. Bulygin, <a href="http://arxiv.org/abs/0810.1619">Towards a Better Understanding of the Semigroup Tree</a>, arXiv:0810.1619 [math.CO], 2008; <a href="http://dx.doi.org/10.1007/s0023300991758">Semigroup Forum 79 (2009) 561574</a>.
%H M. BrasAmoros and A. de Mier, <a href="http://arxiv.org/abs/math/0612634">Representation of Numerical Semigroups by Dyck Paths</a>, arXiv:math/0612634 [math.CO], 2006; <a href="http://dx.doi.org/10.1007/s0023300707177">Semigroup Forum 75 (2007) 676681</a>.
%H Maria BrasAmorós, J FernándezGonzález, <a href="https://arxiv.org/abs/1607.01545">Computation of numerical semigroups by means of seeds</a>, arXiv preprint arXiv:1607.01545 [math.CO], 2016.
%H Shalom Eliahou, Jean Fromentin, <a href="http://images.math.cnrs.fr/SemigroupesnumeriquesetnombredorI.html">Semigroupes Numériques et Nombre d’or</a>, Images des Mathématiques, CNRS, 2018. In French.
%H Sergi Elizalde, <a href="http://arxiv.org/abs/0905.0489">Improved bounds on the number of numerical semigroups of a given genus</a>, arXiv:0905.0489 [math.CO], 2009. [Maria BrasAmoros, Sep 01 2009]
%H G. Failla, C. Peterson, R. Utano, <a href="http://dx.doi.org/10.1007/s0023301596908">Algorithms and basic asymptotics for generalized numerical semigroups in N^d</a>, Semigroup Forum, Jan 31 2015, DOI 10.1007/s0023301596908.
%H S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/">Monoids of natural numbers</a>
%H Jean Fromentin, <a href="http://arxiv.org/abs/1305.3831">Exploring the tree of numerical semigroups</a>, arXiv:1305.3831 [math.CO], 20132015. See <a href="https://hal.archivesouvertes.fr/hal00823339">also</a>, hal00823339.
%H Jean Fromentin, Shalom Eliahou, <a href="http://images.math.cnrs.fr/SemigroupesnumeriquesetnombredorII.html">Semigroupes numériques et nombre d’or (II)</a>, (in French), Images des Mathématiques, CNRS, 2018.
%H Florent Hivert, <a href="https://dx.doi.org/10.1145/3115936.3115938">High Performance Computing Experiments in Enumerative and Algebraic Combinatorics</a>, PASCO 2017 Proceedings of the International Workshop on Parallel Symbolic Computation, Article No. 2.
%H Trevor Hyde, <a href="https://mathoverflow.net/questions/310210/referenceforkakutaniresultonpowersumbasesofsymmetricfunctions">Math Overflow post on reference for result mentioned in Zagier's email.</a>
%H Nathan Kaplan, <a href="https://arxiv.org/abs/1707.02551">Counting Numerical Semigroups</a>, arXiv:1707.02551 [math.CO], 2017. Also Amer. Math. Monthly, 124 (2017), 862875.
%H Jiryo Komeda, <a href="http://dx.doi.org/10.1007/PL00005972">NonWeierstrass numerical semigroups</a>. Semigroup Forum 57 (1998), no. 2, 157185. [Maria BrasAmoros, Sep 01 2009]
%H Nivaldo Medeiros, <a href="http://www.impa.br/~nivaldo/algebra/semigroups/index.html">Numerical Semigroups</a>
%H Alex Zhai, <a href="http://arxiv.org/abs/1111.3142">Fibonaccilike growth of numerical semigroups of a given genus</a>, arXiv:1111.3142 [math.CO], 2011.
%H Y. Zhao, <a href="http://dx.doi.org/10.1007/s0023300991909">Constructing numerical semigroups of a given genus</a>, Semigroup Forum 80 (2010) 242254
%H <a href="/index/Se#semigroups">Index entries for sequences related to semigroups</a>
%F Conjectures: A) a(n) >= a(n1)+a(n2); B) a(n)/(a(n1)+a(n2)) approaches 1; C) a(n)/a(n1) approaches the golden ratio. Conjectures B and C have been proved by Zhai (2011).  Maria BrasAmoros (maria.bras(AT)gmail.com), Oct 24 2007, corrected Aug 31 2009
%e G.f. = x + 2*x^2 + 4*x^3 + 7*x^4 + 12*x^5 + 23*x^6 + 39*x^7 + 67*x^8 + ...
%e a(1)=1 because the unique numerical semigroup with genus 1 is N \ {1}
%Y Row sums of A199711. [corrected by _Jonathan Sondow_, Nov 05 2017]
%K nonn,nice
%O 0,3
%A Don Zagier (don.zagier(AT)mpimbonn.mpg.de), Apr 11 1994
%E The terms from a(17) to a(52) were contributed (in the context of semigroups) by Maria BrasAmoros (maria.bras(AT)gmail.com), Oct 24 2007. The computations were done with the help of Jordi Funollet and Josep M. Mondelo.
%E Terms a(53)a(60) are taken from the Fromentin (2013) paper.  _N. J. A. Sloane_, Sep 05 2013
%E Entry revised by _N. J. A. Sloane_, Aug 31 2009, Sep 02 2009, Feb 07 2012
