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%I A007323 M1064
%S A007323 1,2,4,7,12,23,39,67,118,204,343,592,1001,1693,2857,4806,8045,13467,22464,
%T A007323 37396,62194,103246,170963,282828,467224,770832,1270267,2091030,3437839,
%U A007323 5646773,9266788,15195070,24896206,40761087,66687201,109032500,178158289
%N A007323 Number of bases for symmetric functions of n variables - see Comments 
               lines for precise definition.
%C A007323 Also the number of semigroups of "genus" n.
%C A007323 From Don Zagier's email of Apr 11 1994: (Start)
%C A007323 Given n, one knows that the field of symmetric functions in n variables
%C A007323 a_1,...,a_n is the field Q(sigma_1,...,sigma_n), where
%C A007323 sigma_i is the i-th elementary symmetric polynomial. Here
%C A007323 one has no choice, because sigma_i=0 for i>n and fewer
%C A007323 than n sigma's would not suffice. But, by Newton's formulas,
%C A007323 the field is also given as Q(s_1,...,s_n) where s_i is the
%C A007323 i-th power sum, and now one can ask whether some other sequence
%C A007323 s_{j_1},...,s_{j_n} (0<j_1<...<j_n) also works. For n=1 the
%C A007323 only possibility is clearly s_1, since Q(s_i) = Q(a^i) does
%C A007323 not coincide with Q(a) for i>1, but for n=2 one has two
%C A007323 possibilities Q(s_1,s_2) or Q(s_1,s_3), since from s_1=a+b
%C A007323 and s_3=a^3+b^3 one can reconstruct s_2 = (s_1^3+2s_3)/3s_1.
%C A007323 Similarly, for n=3 one has the possibilities (123), (124),
%C A007323 (125), and (135) (the formula in the last case is
%C A007323 s_2 = (s_1^5+5s_1^2s_3-6s_5)/5(s_1^3-s_3); one can find the
%C A007323 corresponding formulas in the other cases easily) and for
%C A007323 n=4 there are 7: 1234, 1235, 1236, 1237, 1245, 1247, and 1357.
%C A007323 A theorem of Kakutani (I do not know a reference) says that the sequences
%C A007323 which occur are exactly the finite subsets of N whose complements
%C A007323 are additive semigroups (for instance, the complement of {1,2,4,7}
%C A007323 is 3,5,6,8,9,..., which is an closed under addition). This is
%C A007323 a really beautiful theorem. I wrote a simple program to count the sets
%C A007323 of cardinality n which have the property in question for n = 1, ..., 
               16. (End)
%C A007323 Occurs in Blanco, Justo Puerto, p.6, with a(0) = 1 prepended, as Table 
               1: "Number of numerical semigroups with given gender" where by "gender" 
               they may mean "Genus." [From Jonathan Vos Post (jvospost3(AT)gmail.com), 
               Jan 11 2009]
%D A007323 M. Bras-Amoros, Fibonacci-Like Behavior of the Number of Numerical Semigroups 
               of a Given Genus, Semigroup Forum, 76 (2008), 379-384..
%D A007323 M. Bras-Amoros, Bounds on the Number of Numerical Semigroups of a Given 
               Genus, Journal of Pure and Applied Algebra, Elsevier, vol. 213, n. 
               6, pp. 997-1001, June 2009. ISSN: 0022-4049. arXiv:0802.2175.
%D A007323 M. Bras-Amoros and S. Bulygin: Towards a Better Understanding of the 
               Semigroup Tree. Semigroup Forum, Springer. Accepted. ISSN: 0037-1912. 
               arXiv:0810.1619.
%D A007323 M. Bras-Amoros and A. de Mier, Representation of Numerical Semigroups 
               by Dyck Paths, Semigroup Forum. (Volume, pages, year?) arXiv:math/
               0612634.
%D A007323 Sergi Elizalde, Improved bounds on the number of numerical semigroups 
               of a given genus, http://arxiv.org/abs/0905.0489 [From Maria Bras-Amoros, 
               Sep 01 2009]
%D A007323 Jiryo Komeda, Non-Weierstrass numerical semigroups. Semigroup Forum 57 
               (1998), no. 2, 157-185. [From Maria Bras-Amoros, Sep 01 2009]
%D A007323 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A007323 Maria Bras-Amoros, <a href="b007323.txt">Table of n, a(n) for n = 1..50</
               a>
%H A007323 Maria Bras-Amoros, <a href="http://crises-deim.urv.cat/~mbras/">Home 
               Page</a> [Has many of these references]
%H A007323 Victor Blanco, Justo Puerto, <a href="http://arxiv.org/abs/0901.1228">
               Computing the number of numerical semigroups using generating functions</
               a>, Jan 9, 2009. [From Jonathan Vos Post (jvospost3(AT)gmail.com), 
               Jan 11 2009]
%H A007323 S. R. Finch, <a href="http://algo.inria.fr/bsolve/">Monoids of natural 
               numbers</a>
%H A007323 Nivaldo Medeiros, <a href="http://www.impa.br/~nivaldo/algebra/semigroups/
               index.html">Numerical Semigroups</a>
%H A007323 <a href="Sindx_Se.html#semigroups">Index entries for sequences related 
               to semigroups</a>
%F A007323 Comments from Maria Bras-Amoros (maria.bras(AT)gmail.com), Oct 24 2007, 
               corrected Aug 31 2009: Conjectures: A) a(n) >= a(n-1)+a(n-2); B) 
               a(n)/(a(n-1)+a(n-2)) approaches 1; C) a(n)/a(n-1) approaches the 
               golden ratio.
%e A007323 a(1)=1 because the unique numerical semigroup with genus 1 is N \ {1}
%Y A007323 Sequence in context: A072641 A135360 A082548 this_sequence A099604 A026790 
               A054165
%Y A007323 Adjacent sequences: A007320 A007321 A007322 this_sequence A007324 A007325 
               A007326
%K A007323 nonn,nice
%O A007323 1,2
%A A007323 Don Zagier (don.zagier(AT)mpim-bonn.mpg.de), Apr 11 1994
%E A007323 The terms from a(17) onwards were contributed (in the context of semigroups) 
               by Maria Bras-Amoros (maria.bras(AT)gmail.com), Oct 24 2007. The 
               computations were done with the help of Jordi Funollet and Josep 
               M. Mondelo.
%E A007323 Entry revised by N. J. A. Sloane, Aug 31 2009 and Sep 02 2009

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