A029726 -id:A029726 - cp's OEIS frontend

A029725 Number of distinct subgroups of alternating group A_n, counting conjugates as distinct.

1, 1, 1, 2, 10, 59, 501, 3786, 48337, 508402, 6469142, 81711572, 2019160542, 31945830446, 749115591093, 15230426073946, 617313283787616, 13472047923890487

Offset: 0

N. J. A. Sloane

nonn

L. Naughton, 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
N. J. A. Sloane, Transforms

Cf. A005432. Labeled version of A029726.

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

Exponential transform of A116652. - Christian G. Bower, Feb 23 2006

a(9)-a(13) added by Christian G. Bower, Feb 23 2006 based on Goetz Pfeiffer's web page.

a(14)-a(17) from Vaclav Kotesovec, Jul 21 2022

A116653 Number of connected even permutation groups; conjugacy classes of connected subgroups of the alternating group A_n; atomic species based on even permutation groups.

Original entry on oeis.org

0, 1, 0, 1, 3, 4, 12, 15, 87, 61, 143, 168, 1254, 1079, 5281, 7619, 56670, 58957

Offset: 0

Christian G. Bower, Feb 23 2006

L. Naughton, G. Pfeiffer, Integer Sequences Realized by the Subgroup Pattern of the Symmetric Group, J. Int. Seq. 16 (2013) #13.5.8
N. J. A. Sloane, Transforms

Unlabeled version of A116652.

Inverse Euler transform of A029726. Define b(n), c(n): b(1)=c(0)=0. b(k)=A116654(k), k>1. c(k)=A029726(k), k>0. a(n) is Dirichlet convolution of b and c.

a(14)-a(17) from Vaclav Kotesovec, Jul 21 2022

A066113 a(n) is the number of conjugacy classes of maximal subgroups of the alternating group A_n.

Original entry on oeis.org

0, 1, 2, 3, 5, 5, 6, 8, 7, 7, 11, 9, 9, 11, 12, 10, 13, 10, 14, 14, 13, 13, 19, 15, 15, 18, 20, 15, 21, 20, 22, 20, 18, 21, 27, 19, 21, 21, 27, 21, 27, 22, 26, 29, 24, 24, 32, 27, 31, 27, 31, 27, 33, 30, 35, 34, 30, 30, 41, 31, 33, 40, 40, 41, 40, 34, 39, 36, 40

Offset: 2

Reiner Martin, Dec 30 2001

nonn

Andrew Howroyd, Table of n, a(n) for n = 2..250
M. W. Liebeck, C. E. Praeger and J. Saxl, A classification of the maximal subgroups of the finite alternating and symmetric groups, J. Algebra 111(1987), no. 2, 365-383; Math. Rev. 89b:20008.

Cf. A029726, A066115, A316435.

List([2..50],i->Length(MaximalSubgroupClassReps(AlternatingGroup(i))));

More terms from Alexander Hulpke, Feb 19 2002

Terms a(51) and beyond from Andrew Howroyd, Jul 02 2018

A070733 Size of largest conjugacy class in A_n, the alternating group on n symbols.

Original entry on oeis.org

1, 1, 1, 4, 20, 90, 630, 3360, 30240, 226800, 2494800, 23950080, 311351040, 3632428800, 54486432000, 747242496000, 12703122432000, 200074178304000, 3801409387776000, 67580611338240000, 1419192838103040000, 28100018194440192000, 646300418472124416000

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 14 2002

nonn

For n > 5, the largest conjugacy class in A_n corresponds to the cycle type (n-2, 2) if n is even, (n-3, 2, 1) if n is odd. - Eric M. Schmidt, Sep 13 2014

Eric M. Schmidt, Table of n, a(n) for n = 1..100

Cf. A059171, A001710, A000702, A029726.

a:=function(n)
local G,CC,SCC,SCC1;
G:=AlternatingGroup(n);
CC:=ConjugacyClasses(G);;
SCC:=List(CC,Size);
return Maximum(SCC);
end;;  #  W. Edwin Clark, Feb 02 2014

a[n_] := (n!/2) / If[OddQ[n],  n-3, n-2]; a[1] = a[2] = a[3] = 1; a[4] = 4; a[5] = 20; Array[a, 20] (* Amiram Eldar, Jul 12 2025 *)

a(n) = if(n < 6, [1, 1, 1, 4, 20][n], (n!/2) / if(n % 2,  n-3, n-2)); \\ Amiram Eldar, Jul 12 2025

For n > 5, a(n) = n!/(2(n-2)) if n is even, a(n) = n!/(2(n-3)) if n is odd. - Eric M. Schmidt, Sep 13 2014

Sum_{n>=1} 1/a(n) = 111/10 + 1/e - 3*e. - Amiram Eldar, Jul 12 2025

More terms from Eric M. Schmidt, Sep 13 2014

A116654 Number of atomic species based on even permutation groups that are not substitutions of smaller species.

Original entry on oeis.org

0, 1, 0, 1, 3, 4, 11, 15, 84, 59, 139, 168, 1232, 1079

Offset: 0

Christian G. Bower, Feb 23 2006

Define b(n), c(n): b(1)=c(0)=0. b(k)=a(k), k>1. c(k)=A029726(k), k>0. A116653(n) is Dirichlet convolution of b and c.

cp's OEIS Frontend

A029725 Number of distinct subgroups of alternating group A_n, counting conjugates as distinct.

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A116653 Number of connected even permutation groups; conjugacy classes of connected subgroups of the alternating group A_n; atomic species based on even permutation groups.

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A066113 a(n) is the number of conjugacy classes of maximal subgroups of the alternating group A_n.

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Author

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GAP

Extensions

A070733 Size of largest conjugacy class in A_n, the alternating group on n symbols.

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GAP

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A116654 Number of atomic species based on even permutation groups that are not substitutions of smaller species.

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