A000638 - OEIS

%I M1244 N0477 #83 Dec 23 2023 14:29:05

%S 1,1,2,4,11,19,56,96,296,554,1593,3094,10723,20832,75154,159129,

%T 686165,1466358,7274651

%N Number of permutation groups of degree n; also number of conjugacy classes of subgroups of symmetric group S_n; also number of molecular species of degree n.

%D F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 147.

%D Labelle, Jacques. "Quelques espèces sur les ensembles de petite cardinalité.", Ann. Sc. Math. Québec 9.1 (1985): 31-58.

%D G. Pfeiffer, Counting Transitive Relations, preprint 2004.

%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 H. Decoste, G. Labelle, & J. Labelle, <a href="/A005226/a005226.pdf">Espèces sur les petites cardinalités Tableaux divers</a>, Université du Québec à Montréal (octobre 1988), Unpublished.

%H Justine Falque, <a href="https://hal.archives-ouvertes.fr/hal-02918958/document#page=26">On the enumeration of P-oligomorphic groups</a>, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 25-26.

%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 Jacques Labelle, <a href="/A005226/a005226_1.pdf">Quelques espèces sur les ensembles de petite cardinalité</a>, Ann. Sc. Math. Québec 9.1 (1985): 31-58. (Annotated scanned copy of preprint)

%H A. C. Lunn and J. K. Senior, <a href="http://dx.doi.org/10.1021/j150301a009">Isomerism and Configuration</a>, J. Physical Chem. 33 (7) 1929, 1027-1079.

%H A. C. Lunn and J. K. Senior, <a href="/A000637/a000637.pdf">Isomerism and Configuration</a>, J. Physical Chem. 33 (7) 1929, 1027-1079. [Annotated scan of page 1069 only]

%H L. Naughton and G. Pfeiffer, <a href="http://arxiv.org/abs/1211.1911">Integer Sequences Realized by the Subgroup Pattern of the Symmetric Group</a>, arXiv preprint arXiv:1211.1911 [math.GR], 2012 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Naughton/naughton2.html">J. Int. Seq. 16 (2013) #13.5.8</a>

%H Götz Pfeiffer, <a href="http://schmidt.nuigalway.ie/subgroups">Numbers of subgroups of various families of groups</a>

%H G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Pfeiffer/pfeiffer6.html">Counting Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

%H Colin D. Reid, Simon M. Smith, <a href="https://arxiv.org/abs/2002.11766">Groups acting on trees with Tits' independence property (P)</a>, arXiv:2002.11766 [math.GR], 2020.

%H C. C. Sims, <a href="/A000019/a000019.pdf">Letter to N. J. A. Sloane (no date)</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H Dashiell Stander, Qinan Yu, Honglu Fan, and Stella Biderman, <a href="https://arxiv.org/abs/2312.06581">Grokking Group Multiplication with Cosets</a>, arXiv:2312.06581 [cs.LG], 2023. See footnote, p. 25.

%H G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~algebra~permgroup.en.html">PermGroup</a>

%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>

%F Euler Transform of A005226. Define b(n), c(n), d(n): b(1)=d(1)=0. b(k)=A005227(k), k>1. c(k)=a(k), k>0, d(k)=A005226(k), k>1. d is Dirichlet convolution of b and c. - _Christian G. Bower_, Feb 23 2006

%o (Magma) n := 5; #SubgroupLattice(Sym(n));

%o (GAP)

%o # GAP 4.2

%o Length(ConjugacyClassesSubgroups(SymmetricGroup(n)));

%Y Partial sums of A000637.

%Y Cf. A000001, A000019. Unlabeled version of A005432.

%K nonn,hard,more,nice

%O 0,3

%A _N. J. A. Sloane_

%E a(11) corrected and a(12) added by Goetz Pfeiffer (goetz.pfeiffer(AT)nuigalway.ie), Jan 21 2004

%E Extended to a(18) using Derek Holt's data from A000637. - _N. J. A. Sloane_, Jul 31 2010