A005432 - OEIS

A005432

Number of permutation groups of degree n (or, number of distinct subgroups of symmetric group S_n, counting conjugates as distinct).
(Formerly M1690)

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

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

Table of n, a(n) for n=0..18.

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

FORMULA

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

PROG

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

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

CROSSREFS

Cf. A000001, A000019, A000638.

Sequence in context: A192446 A357582 A218940 * A009422 A057221 A180892

Adjacent sequences: A005429 A005430 A005431 * A005433 A005434 A005435

KEYWORD

nonn,hard,more,nice