A000638 -id:A000638 - cp's OEIS frontend

A173397 Partial sums of A000019.

1, 2, 4, 6, 11, 15, 22, 29, 40, 49, 57, 63, 72, 76, 82, 104, 114, 118, 126, 130, 139, 143, 150, 155, 183, 190, 205, 219, 227, 231, 243, 250, 254, 256, 262, 284, 295, 299, 301, 309, 319, 323, 333, 337, 346, 348, 354, 358, 398, 407, 409, 412, 420, 424, 432, 441

Offset: 1

Jonathan Vos Post, Feb 17 2010

nonn

Partial sums of number of primitive permutation groups of degree n. The subsequence of primes in this partial sum begins: 2, 11, 29, 139, 227, 337, 409, 463, 563, 593, 821, 853, 881 (and other powers include 243). The subsequence of squares in this partial sum begins: 1, 4, 49, 256, 441, 576.

Vaclav Kotesovec, Table of n, a(n) for n = 1..4095

Cf. A000019, A000001, A023675, A023676, A000638, A002106, A005432, A000637.

Cases[Import["https://oeis.org/A000019/b000019.txt", "Table"], {, }][[All, 2]] // Accumulate (* Jean-François Alcover, Jan 03 2020 *)

a(n) = Sum_{i=1..n} A000019(i).

A174511 The number of isomorphism classes of subgroups of the symmetric group S_n.

Original entry on oeis.org

1, 2, 4, 9, 16, 29, 55, 137, 241, 453, 894, 2065, 3845

Offset: 1

W. Edwin Clark, Nov 28 2010

Two subgroups are considered to be isomorphic here if they are isomorphic as abstract groups, not as permutation groups. - N. J. A. Sloane, Nov 28 2010

			a(3) = 4 since S_3 contains up to isomorphism exactly one subgroup of each of the orders 1,2,3,6.

A. Distler and T. Kelsey, The semigroups of order 9 and their automorphism groups, arXiv preprint arXiv:1301.6023 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 19 2013
J. Schmidt, Enumerating all subgroups of the symmetric group.

Cf. A000638, A005432.

a:=[];
for n in [1,2,3,4,5,6,7,8,9,10] do
  G:=SymmetricGroup(n);
  R:=ConjugacyClassesSubgroups(G);
  RR:=ListX(R,Representative);
  T:=[RR[1]];
  for g in RR do
    flag:=false;
    for h in T do
      if IsomorphismGroups(g,h)<>fail then
        flag:=true;
        break;
      fi;
    od;
    if flag=false then Add(T,g); fi;
  od;
  Add(a,Size(T));
od;
Print(a,"\n");

a(11) and a(12) from Stephen A. Silver, Feb 24 2013

a(13) (as calculated by Jack Schmidt) from L. Edson Jeffery, May 26 2013

A343592 Number of symmetry types of digraphs with n nodes.

Original entry on oeis.org

1, 2, 4, 9, 14, 36

Offset: 1

Peter Dolland, Apr 21 2021

The symmetry type of a digraph is determined by its automorphism group. It is a permutation group on the nodes set, and therefore a subgroup of the symmetric group Sn. The total number of these is determined by A000638. But not all of them occur as an automorphism group of a digraph.

			The four symmetry types of the digraphs with 3 nodes are represented by:
1.) {}, the empty graph, has together with the full graph the automorphism group S_3 (as subgroup of S_3) as symmetry type.
2.) {(1,2)} has together with 6 other digraphs the trivial automorphism group {id} as symmetry type. This digraph class is called asymmetric. Their values are given by A051504.
3.) {(1,2),(2,1)} has together with 5 other digraphs the automorphism group containing id and a transposition (so it is C_2 as the subgroup of S_3) as symmetry type.
4.) {(1,2),(2,3),(3,1)} has as the only digraph with three nodes the automorphism group C_3 as symmetry type. As a consequence it has to be self-complementary.
The total of the sizes of the symmetry type classes yields the number of digraphs A000273. Here: 2+7+6+1 = 16 = A000273(3).
Note, that for n > 3 there may be different symmetry types with isomorphic automorphism groups. For n=4 both {(1,2)} and {(1,2),(3,4)} have C_2 as automorphism group, but they are different as permutation group.

Peter Dolland, Digraph Symmetry Tables for n = 3..6
Götz Pfeiffer, Subgroups. [broken link]

Cf. A000273, A000638, A051504, A053763, A000595.

A091070 Number of automorphism groups of partial orders on n points.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 16, 21, 41, 57, 103, 140, 276

Offset: 0

Goetz Pfeiffer (goetz.pfeiffer(AT)nuigalway.ie), Jan 21 2004

			a(3)=3 because of the 5 partial orders on 3 points, 2 have trivial automorphism group, 2 have an automorphism of order 2 and one has the full symmetric group as its automorphism group; thus 3 different (conjugacy classes of) subgroups occur.

G. Pfeiffer, Subgroups.
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

Cf. A000638 (subgroups of the symmetric group), A000112 (partial orders).

A141034 Rank of the unit group of the Burnside ring of the symmetric group on n points.

Original entry on oeis.org

1, 2, 3, 6, 10, 23, 34, 67, 110, 205, 320, 660

Offset: 1

Goetz Pfeiffer (goetz.pfeiffer(AT)nuigalway.ie), Jul 30 2008

Boltje, Robert and Pfeiffer, Goetz, An algorithm for the unit group of the Burnside ring of a finite group. Groups St. Andrews 2005. Vol. 1, pp. 230-236, London Math. Soc. Lecture Note Ser., 339, Cambridge Univ. Press, Cambridge, 2007.

Robert Boltje and Goetz Pfeiffer, An algorithm for the unit group of the Burnside ring of a finite group.
Robert Boltje and Goetz Pfeiffer, An algorithm for the unit group of the Burnside ring of a finite group, arXiv:0808.1232 [math.GR], 2008.

Cf. A000638.

A215651 Number of transformation semigroups acting on n points (counting conjugates as one), i.e., the number of subsemigroups of the full transformation semigroup T_n.

Original entry on oeis.org

1, 2, 8, 283, 132069776

Offset: 0

Attila Egri-Nagy, Aug 19 2012

The semigroup analog of A000638.

We apply the categorical viewpoint and consider the empty set as a semigroup.

James East, Attila Egri-Nagy, James D. Mitchell, Enumerating Transformation Semigroups, Semigroup Forum 95, 109-125 (2017); arXiv: 1403.0274 [math.GR], 2014-2017.

Cf. A000638, A215650.

################################################################################
# GAP 4.5 function calculating the conjugacy classes of a set of subsemigrops.
# (C) 2012 Attila Egri-Nagy www.egri-nagy.hu
# GAP can be obtained from www.gap-system.org
################################################################################
# Input: list of subsemigroups of a transformation semigroup,
#        automorphism group of the semigroup
# Output: list of conjugacy classes
ConjugacyClassesSubsemigroups := function(subsemigroups, G)
local ssg, #subsemigroup
      ccl, #conjugacy class
      ccls; #result: all conjugacy classes
  ccls := [];
  for ssg in subsemigroups do
    #we check whether the subsemigroup is already in a conjugacy class
    if not ForAny(ccls, x -> ssg in x) then
      #conjugating by all group elements
      ccl := DuplicateFreeList(
                     List(G,
                          g -> AsSortedList(List(ssg, t-> t^g))));
      Add(ccls, ccl);
    fi;
  od;
  return ccls;
end;

a(4) moved from a comment by Attila Egri-Nagy, Jan 09 2014 to data by Andrey Zabolotskiy, Mar 25 2021

A347007 Number of cycle types of permutation groups with degree n.

Original entry on oeis.org

1, 1, 2, 4, 11, 19, 55, 93, 285, 535, 1514, 2934

Offset: 0

Peter Dolland, Aug 10 2021

A000638 gives the number of permutation groups of degree n. Each permutation group is assigned a cumulative cycle type resulting from the cycle types of its member permutations.

			The 4 cycle types of the 4 permutation groups with degree 3 may be represented by arrays of length 3 (the number of partitions of 3, A000041(3)), indicating the quantity of member permutations, whose cycle type yields a specific partition of n. The partitions are listed in graded lexicographical ordering (see A193073), here (1^3), (2,1), (3):
   1. [1, 0, 0]
   2. [1, 1, 0]
   3. [1, 0, 2]
   4. [1, 3, 2]
The cycle types belong to the permutation groups {id}, C2, C3, and S3 (all subgroups of S3).
Note: For degree n < 6 all permutation groups have different cycle types, so a(n) = A000638(n). For n = 6 there are exactly two permutation groups with the same cycle type (namely [1, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0], both groups isomorphic with C2^2), so a(6) = 55 = A000638(6) - 1.

Peter Dolland, Cycle types shared by more than one permutation group of degree n = 6..10

Cf. A000638.

# GAP 4.11.1
n := 9;;
G := SymmetricGroup(n);
cc := ConjugacyClasses(G);;
sub := ConjugacyClassesSubgroups(G);;
rep := List(sub, Representative);;
ctlst := List( rep, x-> List( cc, c-> Size( Intersection( x, c))));;
Size( AsDuplicateFreeList( ctlst));

A091071 Number of normalizers of subgroups of the symmetric group on n points.

Original entry on oeis.org

1, 1, 1, 2, 4, 5, 12, 19, 42, 72, 127, 196, 500

Offset: 0

Goetz Pfeiffer (goetz.pfeiffer(AT)nuigalway.ie), Jan 21 2004

			a(3)=2 because of the 4 (conjugacy classes of) subgroups of Sym(3) only 2 (Sym(2) and Sym(3)) are normalizers of subgroups.

G. Pfeiffer, Counting Transitive Relations, preprint, 2004.

G. Pfeiffer, Subgroups.
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

Cf. A000638 (subgroups of Sym(n)), A091070 (stabilizers of partial orders).

A174201 Partial sums of A141034.

Original entry on oeis.org

1, 3, 6, 12, 22, 45, 79, 146, 256, 461, 781, 1441

Offset: 1

Jonathan Vos Post, Mar 11 2010

Partial sums of rank of the unit group of the Burnside ring of the symmetric group on n points. The underlying sequence includes the primes 2, 3, 23, 67; the subsequence of prime values in this partial sum begins: 3, 79, 461.

Cf. A000638, A141034.

a(n) = SUM[i=1..n] A141034(i).

A269890 Number of conjugacy classes of subgroups of the hyperoctahedral group.

Original entry on oeis.org

2, 8, 33, 193, 953, 7440, 55200, 627187, 7510549

Offset: 1

Martin Rubey, Mar 07 2016

Cf. A000638.

Length(ConjugacyClassesSubgroups(CoxeterGroup( "B", n )));

[#Subgroups(CoxeterGroup("B" cat IntegerToString(n))) : n in [1..9]];  // Robin Visser, Aug 09 2023

a(6)-a(9) from Robin Visser, Aug 09 2023

cp's OEIS Frontend

A173397 Partial sums of A000019.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A174511 The number of isomorphism classes of subgroups of the symmetric group S_n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Extensions

A343592 Number of symmetry types of digraphs with n nodes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A091070 Number of automorphism groups of partial orders on n points.

Views

Author

Keywords

Examples

Links

Crossrefs

A141034 Rank of the unit group of the Burnside ring of the symmetric group on n points.

Views

Author

Keywords

References

Links

Crossrefs

A215651 Number of transformation semigroups acting on n points (counting conjugates as one), i.e., the number of subsemigroups of the full transformation semigroup T_n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Extensions

A347007 Number of cycle types of permutation groups with degree n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

A091071 Number of normalizers of subgroups of the symmetric group on n points.

Views

Author

Keywords

Examples

References

Links

Crossrefs

A174201 Partial sums of A141034.

Views

Author

Keywords

Comments

Crossrefs

Formula

A269890 Number of conjugacy classes of subgroups of the hyperoctahedral group.