A005226 Number of atomic species of degree n; also number of connected permutation groups of degree n.

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

Offset: 0

Author

Simon Plouffe

Keywords

• nonn
• more
• hard

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

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& J. Labelle, Espèces sur les petites cardinalités Tableaux divers, Université du Québec à Montréal (octobre 1988), Unpublished.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

Crossrefs

Cf. A005227. Unlabeled version of A116655.

Programs

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 *)

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

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