A005432 Number of permutation groups of degree n (or, number of distinct subgroups of symmetric group S_n, counting conjugates as distinct).

1, 1, 2, 6, 30, 156, 1455, 11300, 151221, 1694723, 29594446, 404126228, 10594925360, 175238308453, 5651774693595, 117053117995400, 5320744503742316, 125889331236297288, 7598016157515302757

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• hard
• more
• nice

Comments

Labeled version of A000638.

L. Pyber shows c^(n^2(1+o(1))) <= a(n) <= d^(n^2(1+o(1))), c=2^(1/16), d=24^(1/6); conjectures lower bound is accurate.

References

• 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 and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Piotr Graczyk, Hideyuki Ishi, Kołodziejek Bartosz, Hélène Massam, Model selection in the space of Gaussian models invariant by symmetry, arXiv:2004.03503 [math.ST], 2020.
• D. F. Holt, Enumerating subgroups of the symmetric group.
• 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]
• J. Labelle and Y. N. Yeh, The relation between Burnside rings and combinatorial species, J. Combin. Theory, A 50 (1989), 269-284.
• L. Naughton and G. Pfeiffer, Integer Sequences Realized by the Subgroup Pattern of the Symmetric Group, J. Int. Seq. 16 (2013) #13.5.8.
• Götz Pfeiffer, Numbers of subgroups of various families of groups
• L. Pyber, Enumerating Finite Groups of Given Order, Ann. Math. 137 (1993), 203-220.
• 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.
• Index entries for sequences related to groups

Crossrefs

Cf. A000001, A000019, A000638.

Programs

GAP

List([2..5],n->Sum( List( ConjugacyClassesSubgroups( SymmetricGroup(n)), Size))); [Alexander Hulpke]

Magma

n := 5; &+[ Length(s):s in SubgroupLattice(Sym(n)) ];

Formula

Exponential transform of A116655. Binomial transform of A116693. - Christian G. Bower, Feb 23 2006

Extensions

a(9) and a(10) from Alexander Hulpke, Dec 03 2004

More terms from a(11) and a(12) added by Christian G. Bower, Feb 23 2006 based on Goetz Pfeiffer's web page.

a(13) added by Liam Naughton, Jun 09 2011

a(14)-a(18) from Holt reference, Wouter Meeussen, Jul 02 2013