A005226 -id:A005226 - cp's OEIS frontend

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.

Original entry on oeis.org

1, 1, 2, 4, 11, 19, 56, 96, 296, 554, 1593, 3094, 10723, 20832, 75154, 159129, 686165, 1466358, 7274651

Offset: 0

N. J. A. Sloane

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).

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& J. Labelle, Espèces sur les petites cardinalités Tableaux divers, Université du Québec à Montréal (octobre 1988), Unpublished.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

Partial sums of A000637.

Cf. A000001, A000019. Unlabeled version of A005432.

# GAP 4.2
Length(ConjugacyClassesSubgroups(SymmetricGroup(n)));

Magma
```
n := 5; #SubgroupLattice(Sym(n));
```

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

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

A000637 Number of fixed-point-free permutation groups of degree n.

Original entry on oeis.org

1, 0, 1, 2, 7, 8, 37, 40, 200, 258, 1039, 1501, 7629, 10109, 54322, 83975, 527036, 780193, 5808293

Offset: 0

N. J. A. Sloane

a(1) = 0 since the trivial group of degree 1 has a fixed point. One could also argue that one should set a(1) = 1 by convention.

G. Butler and J. McKay, The transitive groups of degree up to eleven, Comm. Algebra, 11 (1983), 863-911.
D. Holt, Enumerating subgroups of the symmetric group. 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.
A. Hulpke, Konstruktion transitiver Permutationsgruppen, Dissertation, RWTH Aachen, 1996.
A. Hulpke, Constructing transitive permutation groups, J. Symbolic Comput. 39 (2005), 1-30.
A. Hulpke, Constructing Transitive Permutation Groups, in preparation
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).

G. Butler and J. McKay, The transitive groups of degree up to eleven, Comm. Algebra, 11 (1983), 863-911. [Annotated scanned copy]
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]
A. Hulpke, Transitive groups of small degree
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]
C. C. Sims, Letter to N. J. A. Sloane (no date)
Index entries for sequences related to groups

Cf. A000001, A000019, A000638, A002106, A005432, A005226.

Cf. A000019, A002106. Unlabeled version of A116693.

a(n) = A000638(n) - A000638(n-1). - Christian G. Bower, Feb 23 2006

More terms from Alexander Hulpke
a(2) and a(10) corrected, a(11) and a(12) added by Christian G. Bower, Feb 23 2006
Terms a(13)-a(18) were computed by Derek Holt and contributed by Alexander Hulpke, Jul 30 2010, who comments that he has verified the terms up through a(16).
Edited by N. J. A. Sloane, Jul 30 2010, at the suggestion of Michael Somos

A005227 Number of atomic species of degree n which are not nontrivial substitutions.

Original entry on oeis.org

0, 1, 1, 2, 4, 6, 19, 20, 111, 116, 567, 641, 4718, 4772, 35489, 46012, 389277, 487408, 4616580

Offset: 0

N. J. A. Sloane

Labelle, Jacques. 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).

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.

Cf. A005226.

Define b(n), c(n), d(n): b(1)=d(1)=0. b(k)=a(k), k>1. c(k)=A000638(k), k>0, d(k)=A005226(k), k>1. d is Dirichlet convolution of b and c. - Christian G. Bower, Feb 23 2006

a(8), a(11) corrected and a(12) added by Christian G. Bower, Feb 23 2006 based on Goetz Pfeiffer's edit to A000638

a(13)-a(18) from Vaclav Kotesovec, Jul 18 2022

A007650 Number of set-like atomic species of degree n.

Original entry on oeis.org

0, 1, 1, 1, 3, 1, 6, 1, 10, 4, 12, 1, 33, 1, 29, 13, 64, 1, 100, 1, 156, 30, 187, 1, 443, 10, 476, 78, 877, 1, 1326, 1, 2098, 188, 2745, 36, 5203, 1, 6408, 477, 11084, 1, 15687, 1, 24709, 1241, 33249, 1, 57432, 27, 74529, 2746, 120984, 1, 168668, 194, 264075, 6409, 356624, 1, 579893, 1, 768857, 14898, 1214452, 483, 1669060, 1

Offset: 0

Simon Plouffe

nonn

G. Labelle and P. Leroux, Identities and enumeration: weighting connected components, Abstracts Amer. Math. Soc., 15 (1994), Meeting #896.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Joerg Arndt, Table of n, a(n) for n = 0..100
S. Eberhart & N. J. A. Sloane, Correspondence, 1977
G. Labelle and P. Leroux, Identities and enumeration: weighting connected components, Abstracts Amer. Math. Soc., 15 (1994), Meeting #896. (Annotated scanned copy)
G. Labelle and P. Leroux, An extension of the exponential formula in enumerative combinatorics, The Electronic Journal of Combinatorics, Volume 3, Issue 2 (1996) (The Foata Festschrift volume), Research Paper #R12.
N. J. A. Sloane, Transforms.

Cf. A005226, A007649.

NN = 66;  va = Array[0&, NN]; va[[1]] = 0; va[[2]] = 1; vm = Array[0&, NN]; vm[[1]] = 1; vm[[2]] = 1; For[n = 2, n <= NN - 1, n++, va[[n + 1]] = DivisorSum[n , vm[[#+1]]&]; vm[[n+1]] = 1/n*Sum[DivisorSum[k, #*va[[#+1]] &]*vm[[n-k+1]], {k, 1, n}]]; va (* Jean-François Alcover, Dec 01 2015, adapted from Joerg Arndt's PARI script in A007649 *)

PARI
```
/* see A007649 */
```

Inverse Euler Transform of A007649. Define c(n): c(0)=0. c(k)=A007649(k), k>0. a=MOEBIUSi(c)-c. - Christian G. Bower, Feb 23 2006

Added more terms, Joerg Arndt, Jul 30 2012

cp's OEIS Frontend

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.

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A000637 Number of fixed-point-free permutation groups of degree n.

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A005227 Number of atomic species of degree n which are not nontrivial substitutions.

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A007650 Number of set-like atomic species of degree n.

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Mathematica

PARI

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A116655 Number of connected subgroups of the symmetric group S_n.

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