A116655 -id:A116655

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A005432

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

+10
16

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.

KEYWORD

nonn,hard,more,nice

AUTHOR

N. J. A. Sloane, Simon Plouffe

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

STATUS

approved

A005226

Number of atomic species of degree n; also number of connected permutation groups of degree n.
(Formerly M1563)

+10
6

0, 1, 1, 2, 6, 6, 27, 20, 130, 124, 598, 641, 4850, 4772, 35625, 46074, 389839, 487408, 4617554

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

OFFSET

0,4

COMMENTS

An atomic species is one that is not the product of smaller species. - Christian G. Bower, Feb 23 2006

A permutation group is connected if it is not the direct product of smaller permutation groups. - Christian G. Bower, Feb 23 2006

REFERENCES

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

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

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.

H. Decoste, G. Labelle & J. Labelle, Espèces sur les petites cardinalités Tableaux divers, Université du Québec à Montréal (octobre 1988), Unpublished.

Jacques Labelle, Quelques espèces sur les ensembles de petite cardinalité, Ann. Sc. Math. Québec 9.1 (1985): 31-58. (Annotated scanned copy of preprint)

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, arXiv:1211.1911 [math.GR], 2012-2013 and J. Int. Seq. 16 (2013) #13.5.8.

N. J. A. Sloane, Transforms

FORMULA

Inverse Euler transform of A000638. Define b(n), c(n), d(): b(1)=d(1)=0. b(k)=A005227(k), k>1. c(k)=A000638(k), k>0. d(k)=a(k), k>1. d is Dirichlet convolution of b and c. - Christian G. Bower, Feb 23 2006

MATHEMATICA

A000638 = Cases[Import["https://oeis.org/A000638/b000638.txt", "Table"], {_, _}][[All, 2]];

(* EulerInvTransform is defined in A022562 *)

{0} ~Join~ EulerInvTransform[A000638 // Rest] (* Jean-François Alcover, Dec 03 2019, updated Mar 17 2020 *)

CROSSREFS

Cf. A005227. Unlabeled version of A116655.

KEYWORD

nonn,more,hard

AUTHOR

Simon Plouffe

EXTENSIONS

a(11) corrected and a(12) added by Christian G. Bower, Feb 23 2006 based on Goetz Pfeiffer's edit to A000638.

Could be extended to a(18) now using the new terms for A000637. - N. J. A. Sloane, Jul 30 2010

a(13) from Liam Naughton, Nov 23 2012

a(14)-a(18) from the inverse Euler transform of A000637. - R. J. Mathar, Mar 03 2015

STATUS

approved

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