[seqfan] Re: request for help with a sequence

Neil Sloane njasloane at gmail.com
Mon Dec 7 08:38:34 CET 2020


I'm not sure that A002106 needs any clarification.  It is by far the most
natural version.  It is analogous to any kind of counting problem (trees,
for example). The natural version is the unlabeled case.

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com



On Sun, Dec 6, 2020 at 11:58 PM Brendan McKay <Brendan.McKay at anu.edu.au>
wrote:

> The difference is that A002106 is counting isomorphism classes
> (conjugacy classes in the symmetric group) whereas John is
> regarding conjugates as distinct.  A002106 should be clarified.
>
> Gordon Royle and Gabriel Verret would be the people to know
> about this.
>
> Brendan.
>
> On 7/12/20 3:11 pm, Neil Sloane wrote:
> > Did you look at A002106, a large entry in the OEIS:
> >
> > %I M1316 N0504
> >
> > %S 1,1,2,5,5,16,7,50,34,45,8,301,9,63,104,1954,10,983,8,1117,164,59,7,
> >
> > %T 25000,211,96,2392,1854,8,5712,12,2801324,162,115,407,121279,11,76,
> >
> > %U 306,315842,10,9491,10,2113,10923,56,6
> >
> > %N Number of transitive permutation groups of degree n.
> >
> > %C It is conjectured that this is the number of Galois groups for
> > irreducible polynomials of order n. (All such Galois groups are
> > transitive.) - _Charles R Greathouse IV_, May 28 2014
> >
> > %D G. Butler and J. McKay, personal communication.
> >
> > %D C. C. Sims, Computational methods in the study of permutation groups,
> > pp. 169-183 of J. Leech, editor, Computational Problems in Abstract
> > Algebra. Pergamon, Oxford, 1970.
> >
> > %D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
> > (includes this sequence).
> >
> > %D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer
> > Sequences, Academic Press, 1995 (includes this sequence).
> >
> > %H G. Butler and J. McKay, <a href="
> > http://dx.doi.org/10.1080/00927878308822884">The transitive groups of
> > degree up to eleven</a>, Comm. Algebra, 11 (1983), 863-911.
> >
> > %H G. Butler and J. McKay, <a href="/A000637/a000637_1.pdf">The
> transitive
> > groups of degree up to eleven</a>, Comm. Algebra, 11 (1983), 863-911.
> > [Annotated scanned copy]
> >
> > %H John J. Cannon and Derek F. Hol, <a href="
> > http://www.warwick.ac.uk/~mareg/download/papers/trans32/trans32.pdf">The
> > transitive permutation groups of degree 32</a>
> >
> > %H F. N. Cole, <a href="
> http://dx.doi.org/10.1090/S0002-9904-1893-00137-7">Note
> > on the substitution groups of six, seven, and eight letters</a>, Bull.
> > Amer. Math. Soc. 2 (1893), 184-190. Gives a(8)=48 instead of 50.
> >
> > %H Computational Algebra Group, <a href="
> > https://magma.maths.usyd.edu.au/magma/releasenotes/2/21/#subsection_10_8
> ">Summary
> > of New Features in Magma V2.21</a>
> >
> > %H J. Conway, A. Hulpke, and J. McKay, <a href="
> > http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.792">On
> > Transitive Permutation Groups</a>, LMS Journal of Computation and
> > Mathematics 1 (1998), pp. 1-8. See especially Appendix A.
> >
> > %H D. Holt, <a href="/A000019/a000019_1.pdf">Enumerating subgroups of the
> > symmetric group</a>, in Computational Group Theory and the Theory of
> > Groups, II, edited by L.-C. Kappe, A. Magidin and R. Morse. AMS
> > Contemporary Mathematics book series, vol. 511, pp. 33-37. [Annotated
> copy]
> >
> > %H Derek Holt and Gordon Royle, <a href="
> https://arxiv.org/abs/1811.09015">A
> > Census of Small Transitive Groups and Vertex-Transitive Graphs</a>,
> > arXiv:1811.09015 [math.CO], 2018.
> >
> > %H A. Hulpke, <a
> > href="http://www.math.colostate.edu/~hulpke/smalldeg.html">Transitive
> > groups of small degree</a>
> >
> > %H A. Hulpke, <a href="
> > http://www.math.colostate.edu/~hulpke/paper/prom.ps.gz">Konstruktion
> > transitiver Permutationsgruppen</a>, Dissertation, RWTH Aachen, 1996.
> >
> > %H A. Hulpke, <a
> > href="http://dx.doi.org/10.1016/j.jsc.2004.08.002">Constructing
> > transitive permutation groups</a>, J. Symbolic Comput. 39 (2005), 1-30.
> >
> > %H E. G. Köhler, F. H. Lutz, <a
> > href="http://arXiv.org/abs/math.CO/0506520">Triangulated
> > manifolds with few vertices: Vertex-transitive triangulations</a>,
> > arXiv:math/0506520 [math.GT], 2005.
> >
> > %H J. Labelle and Y. N. Yeh, <a href="
> > http://dx.doi.org/10.1016/0097-3165(89)90019-8">The relation between
> > Burnside rings and combinatorial species</a>, J. Combin. Theory, A 50
> > (1989), 269-284. See page 280.
> >
> > %H G. A. Miller, <a href="
> http://dx.doi.org/10.1090/S0002-9904-1896-00327-X
> > ">On the lists of all the substitution groups that can be formed with a
> > given number of elements</a>, Bull. Amer. Math. Soc., 2 (1896), 138-145.
> >
> > %H Wikipedia, <a
> > href="http://en.wikipedia.org/wiki/Inverse_Galois_problem">Inverse
> > Galois problem</a>
> >
> > %H <a href="/index/Gre#groups">Index entries for sequences related to
> > groups</a>
> >
> > %H <a href="/index/Cor#core">Index entries for "core" sequences</a>
> >
> > %e a(3)=2: A_3 and S_3.
> >
> > %o (GAP) a:=function(n)
> >
> > %o return Length(AllTransitiveGroups(NrMovedPoints,n));
> >
> > %o end; # _Charles R Greathouse IV_, May 28 2014
> >
> > %Y Cf. A000001, A000019, A177244, A186277.
> >
> > %K nonn,core,hard,more,nice
> >
> > %O 1,3
> >
> > %A _N. J. A. Sloane_
> >
> > %E Corrected and extended to degree 31 by Alexander Hulpke
> > (Alexander.Hulpke(AT)Math.RWTH-Aachen.DE), Aug 15 1996
> >
> > %E Further corrections from Alexander Hulpke, Feb 19 2002
> >
> > %E Degree 32 extended by _Artur Jasinski_, Feb 17 2011
> >
> > %E Extended to degree 47 by _Gabriel Verret_, May 07 2016
> > Best regards
> > Neil
> >
> > Neil J. A. Sloane, President, OEIS Foundation.
> > 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> > Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> > Phone: 732 828 6098; home page: http://NeilSloane.com
> > Email: njasloane at gmail.com
> >
> >
> >
> > On Sun, Dec 6, 2020 at 10:38 PM John Erickson <ericksonjohn at gmail.com>
> > wrote:
> >
> >> Hi All,
> >>
> >> Just joined seqfan list 30 years too late. I never noticed it before
> and in
> >> particular I never noticed that you can request assistance.
> >> I have recently considered the sequence a(n) defined as
> >>
> >> The number of distinct transitive subgroups of S_n, counting conjugates
> as
> >> distinct.
> >>
> >> With the first 13 terms starting at n = 1 given by
> >> 1, 1, 2, 9, 20, 279, 512, 19087, 71602, 636365, 1517042, 321965982,
> >> 240609602
> >>
> >>
> >> transitiveSubgroupsCount := function(n)
> >>      local numTransitiveSubgroups, G, cc, class;
> >>      numTransitiveSubgroups := 0;
> >>      G := SymmetricGroup(n);
> >>      cc:=ConjugacyClassesSubgroups(G);
> >>      for class in cc do
> >>          if  IsTransitive(Representative(class), [1..n]) then
> >>                 numTransitiveSubgroups := numTransitiveSubgroups +
> >> Size(class);
> >>          fi;
> >>      od;
> >> return numTransitiveSubgroups;
> >> end;
> >>
> >> I have proven that A005432(p)-a(p) == 1 (mod p) if p is prime and
> >> based on n<= 13,  I have conjectured that floor(log(A005432(n)/a(n))) <=
> >> (n-1)/2 for n>=1, with equality holding for n>=3 and prime.
> >>
> >> Obviously, with only 13 examples this may be ridiculous so I would like
> >> more data. Can anyone suggest better GAP code that uses less memory?
> It's
> >> my first GAP program as Mathematica was not well suited for this task.
> >>
> >> In general I would appreciate any suggestions. If I am doing something
> >> particularly inept, feel free to let me know. Thanks.
> >>
> >> John Erickson
> >>
> >> --
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> >>
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>
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