A000638 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.
1, 1, 2, 4, 11, 19, 56, 96, 296, 554, 1593, 3094, 10723, 20832, 75154, 159129, 686165, 1466358, 7274651
Offset: 0
References
- F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 147.
- Labelle, Jacques. "Quelques espèces sur les ensembles de petite cardinalité.", Ann. Sc. Math. Québec 9.1 (1985): 31-58.
- G. Pfeiffer, Counting Transitive Relations, preprint 2004.
- 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.
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- H. Decoste, G. Labelle, & J. Labelle, Espèces sur les petites cardinalités Tableaux divers, Université du Québec à Montréal (octobre 1988), Unpublished.
- Justine Falque, On the enumeration of P-oligomorphic groups, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 25-26.
- D. Holt, Enumerating subgroups of the symmetric group, 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]
- Jacques Labelle, Quelques espèces sur les ensembles de petite cardinalité, Ann. Sc. Math. Québec 9.1 (1985): 31-58. (Annotated scanned copy of preprint) published volume
- A. C. Lunn and J. K. Senior, Isomerism and Configuration, J. Physical Chem. 33 (7) 1929, 1027-1079.
- A. C. Lunn and J. K. Senior, Isomerism and Configuration, J. Physical Chem. 33 (7) 1929, 1027-1079. [Annotated scan of page 1069 only]
- L. Naughton and G. Pfeiffer, Integer Sequences Realized by the Subgroup Pattern of the Symmetric Group, arXiv preprint arXiv:1211.1911 [math.GR], 2012 and J. Int. Seq. 16 (2013) #13.5.8
- Götz Pfeiffer, Numbers of subgroups of various families of groups
- G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
- Colin D. Reid, Simon M. Smith, Groups acting on trees with Tits' independence property (P), arXiv:2002.11766 [math.GR], 2020.
- C. C. Sims, Letter to N. J. A. Sloane (no date)
- N. J. A. Sloane, Transforms
- Dashiell Stander, Qinan Yu, Honglu Fan, and Stella Biderman, Grokking Group Multiplication with Cosets, arXiv:2312.06581 [cs.LG], 2023. See footnote, p. 25.
- G. Xiao, PermGroup
- Index entries for sequences related to groups
Programs
-
GAP
# GAP 4.2 Length(ConjugacyClassesSubgroups(SymmetricGroup(n)));
-
Magma
n := 5; #SubgroupLattice(Sym(n));
Formula
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
Extensions
a(11) corrected and a(12) added by Goetz Pfeiffer (goetz.pfeiffer(AT)nuigalway.ie), Jan 21 2004
Extended to a(18) using Derek Holt's data from A000637. - N. J. A. Sloane, Jul 31 2010