A005432 -id:A005432 - 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

A243748 Irregular triangle read by rows where T(n,k) is the number of subgroups of order d of the symmetric group S_n, where d is the k-th divisor of n!.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 9, 4, 7, 4, 3, 1, 1, 1, 25, 10, 35, 6, 30, 15, 6, 15, 0, 6, 5, 0, 0, 1, 1, 1, 75, 40, 255, 36, 280, 255, 10, 36, 150, 0, 45, 50, 36, 90, 0, 30, 0, 0, 30, 12, 10, 0, 0, 12, 0, 0, 0, 1, 1, 1, 231, 175, 1295, 126, 1645, 120, 1575, 70, 378, 1715, 120, 0, 315, 350, 378, 120, 1435, 0, 0, 0, 245, 126, 120, 0

Offset: 1

R. J. Mathar, Jun 09 2014

The columns skip counting the subgroups of S_n with orders d that do not divide the order of S_n, n!, because such subgroups cannot exist. This is just a reduction of columns in the triangle by omitting a large number of zeros.

			There are T(3,2)=3 subgroups of S_3 of order 2, namely the groups generated by the permutations (1,2), (1,3) or (2,3).
Triangle begins:
  1;
  1,1;
  1,3,1,1;
  1,9,4,7,4,3,1,1;
  1,25,10,35,6,30,15,6,15,0,6,5,0,0,1,1;
  ...

Cf. A005432 (row sums), A001189 (column d=2), A027423 (row lengths), A218913, A277566, A284210.

# GAP 4
LoadPackage("SONATA") ;;
Print("\n") ;
N := Factorial(7) ;; # adjusted to the maximum n below
subS := EmptyPlist(N) ;;
for n in [1..7] do
    for e in [1..N] do
        subS[e] := 0 ;
    od;
    g := SymmetricGroup(n) ;
    sg := Size(g) ;
    alls := Subgroups(g) ;
    for s in alls do
        o := Size(s) ;
        if o <= N then
            subS[o] := subS[o]+1 ;;
        fi;
    od ;
    for d in [1..N] do
        if ( sg mod d ) = 0 then
            Print(subS[d],",") ;
        fi;
    od;
    Print("\n") ;
od;

Edited by Peter Munn, Mar 06 2025

A062297 Number of distinct Abelian subgroups of the symmetric group S_n.

Original entry on oeis.org

1, 2, 5, 21, 87, 612, 3649, 35515, 289927, 377118, 36947363, 657510251, 7736272845

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 02 2001

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

Cf. A005432, A051625.

# GAP 4.4
LoadPackage("sonata");;
for n in [2,3,4,5,6,7,8,9,10] do
    p := SymmetricGroup(n) ;;
    o := Order(p);
    s := Subgroups(p);
    f := Filtered(s, g -> IsAbelian(g));
    a := Size(f);
    Print(a," ") ;
od; # R. J. Mathar, May 24 2013

a(9)-a(13) added by Liam Naughton

A215650 Number of transformation semigroups acting on n points (counting conjugates as distinct); also the number of subsemigroups of the full transformation semigroup T_n.

Original entry on oeis.org

1, 2, 10, 1299, 3161965550

Offset: 0

Attila Egri-Nagy, Aug 19 2012

The semigroup analog of A005432.
We apply the categorical viewpoint and consider the empty set as a semigroup.
The first 4 terms can be calculated by brute force search (see attached program).

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. A005432, A215651.

################################################################################
# GAP 4.5 function implementing a brute force search for submagmas of a magma.
# (C) 2012 Attila Egri-Nagy www.egri-nagy.hu
# GAP can be obtained from www.gap-system.org
################################################################################
# The function goes through all the subsets of the given magma (groups,
# semigroups) and checks whether they form a magma or not.
# If yes, then the submagma is collected.
# The function returns the list of all (nonempty) submagmas.
BruteForceSubMagmaSearch := function(M)
local bitlist, #the characteristic function of a subset
      i, #an integer to index through the bitlist
      n, #size of the input magma
      elms, #elements of the magma
      gens, #generator set of a submagma
      submagmas, #the submagmas
      duplicates, #for counting how many times we encounter the same submagma
      nonsubmagmas; #counting how many subsets are not submagmas
      # duplicates + nonsubmagmas = 2^n-1
  n := Size(M);
  submagmas := [];
  elms := AsList(M);
  duplicates := 0;
  nonsubmagmas := 0;
  bitlist := BlistList([1..n],[1]); #we start with the first element, the
  #empty set can be added afterwards, if the magma's definition allows it
  repeat
    #constructing a generator set based on the bitlist##########################
    gens := [];
    Perform([1..n],function(x) if bitlist[x] then Add(gens, elms[x]);fi;end);
    #checking whether it is a submagma
if Size(gens) = Size(Magma(gens)) then
      if gens in submagmas then
        duplicates := duplicates + 1;
      else
        AddSet(submagmas,gens);
      fi;
    else
      nonsubmagmas := nonsubmagmas + 1;
    fi;
    #binary +1 applied to bitlist###############################################
    i := 1;
    while (i<=n) and (bitlist[i]) do
      bitlist[i] := false;
      i := i + 1;
    od;
    if i <= n then bitlist[i] := true;fi;
    ############################################################################
  until SizeBlist(bitlist) = 0;
  Print("#I Submagmas:", Size(submagmas),
        " Duplicates:", duplicates,
        " Nonsubmagmas:", nonsubmagmas,"\n");
  return submagmas;
end;

a(4) moved from comments to data by Andrey Zabolotskiy, Mar 25 2021

A337015 Number of distinct transitive subgroups of S_n, counting conjugates as distinct.

Original entry on oeis.org

1, 1, 2, 9, 20, 279, 512, 19087, 71602, 636365, 1517042, 321965982, 240609602, 8809543877, 144729615032, 26818608209252, 6603755558402, 2737593592637477

Offset: 1

John Erickson and Alexander Hulpke, Nov 21 2020

This sequence is the labeled version of A002106. I have proven that A005432(p)-a(p) == 1 (mod p) if p is prime. Based on n<= 18,
I have conjectured that log(A005432(n)/a(n)) > (n-1)/2 for n prime and log(A005432(n)/a(n)) < (n-1)/2 for n composite.
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.

			For n = 4 the following 9 subgroups of S_4 are transitive:
Group( [ (1,4)(2,3), (1,3)(2,4) ] )
Group( [ (1,3,2,4), (1,2)(3,4) ] )
Group( [ (1,4,3,2), (1,3)(2,4) ] )
Group( [ (1,2,4,3), (1,4)(2,3) ] )
Group( [ (1,4)(2,3), (1,3)(2,4), (3,4) ] )
Group( [ (1,2)(3,4), (1,3)(2,4), (1,4) ] )
Group( [ (1,2)(3,4), (1,4)(2,3), (2,4) ] )
Group( [ (1,4)(2,3), (1,3)(2,4), (2,4,3) ] )
Group( [ (1,4)(2,3), (1,3)(2,4), (2,4,3), (3,4) ] )

John Erickson, COUNTING TRANSITIVE SUBGROUPS OF Sn
L. Pyber, Enumerating Finite Groups of Given Order, Ann. Math. 137 (1993), 203-220.

Cf. A005432, A002106.

NrTransSubSn:=function(n)
local s,cnt,i,u,no;
  s:=SymmetricGroup(n);
  cnt:=0;
  for i in [1..NrTransitiveGroups(n)] do
    u:=TransitiveGroup(n,i);
    no:=Normalizer(s,u);
    cnt:=cnt+IndexNC(s,no);
    Print("Class ",i,", found ",IndexNC(s,no)," new, total: ",cnt,"\n");
  od;
  return cnt;
end; # Alexander Hulpke

A386593 Number of sub-relation-algebras of Re(n), the collection of all binary relations over {1,2,...,n}.

Original entry on oeis.org

1, 2, 6, 30, 124

Offset: 1

Jeremy F. Alm, Jul 26 2025

The first four terms are not that difficult to verify. In fact, for n < 5 the subalgebra lattice of Re(n) is the dual of the subgroup lattice of S_n. Hence a(n) = A005432(n) for n < 5.

Relation algebras are closed under union, intersection, complement, composition, inverse, and identity.

Bjarni Jónsson, Maximal algebras of binary relations. In: Contributions to Group Theory, Contemporary Mathematics, vol. 33, pp. 299-307. Amer. Math. Soc., Providence (1984).

Cf. A005432.

cp's OEIS Frontend

A173397 Partial sums of A000019.

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

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A243748 Irregular triangle read by rows where T(n,k) is the number of subgroups of order d of the symmetric group S_n, where d is the k-th divisor of n!.

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A062297 Number of distinct Abelian subgroups of the symmetric group S_n.

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A215650 Number of transformation semigroups acting on n points (counting conjugates as distinct); also the number of subsemigroups of the full transformation semigroup T_n.

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Programs

GAP

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A337015 Number of distinct transitive subgroups of S_n, counting conjugates as distinct.

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Programs

GAP

A386593 Number of sub-relation-algebras of Re(n), the collection of all binary relations over {1,2,...,n}.

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