A218913 -id:A218913 - cp's OEIS frontend

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!.

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

A277566 Irregular table by rows: Orders of subgroups of the symmetric group S_n.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 6, 1, 2, 3, 4, 6, 8, 12, 24, 1, 2, 3, 4, 5, 6, 8, 10, 12, 20, 24, 60, 120, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 36, 48, 60, 72, 120, 360, 720, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 21, 24, 36, 40, 42, 48, 60, 72, 120, 144, 168, 240, 360, 720, 2520, 5040, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 30, 32, 36, 40, 42, 48, 56, 60, 64, 72, 96, 120, 128, 144, 168, 180, 192, 240, 288, 336, 360, 384, 576, 720, 1152, 1344, 1440, 2520, 5040, 20160, 40320

Offset: 1

Charles R Greathouse IV, Nov 28 2016

Each row begins with 1 (the order of the trivial group) and ends with n! (the order of S_n itself).

			1;
1, 2;
1, 2, 3, 6;
1, 2, 3, 4, 6, 8, 12, 24;
1, 2, 3, 4, 5, 6, 8, 10, 12, 20, 24, 60, 120;
1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 36, 48, 60, 72, 120, 360, 720;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 21, 24, 36, 40, 42, 48, 60, 72, 120, 144, 168, 240, 360, 720, 2520, 5040;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 30, 32, 36, 40, 42, 48, 56, 60, 64, 72, 96, 120, 128, 144, 168, 180, 192, 240, 288, 336, 360, 384, 576, 720, 1152, 1344, 1440, 2520, 5040, 20160, 40320;

Charles R Greathouse IV, Rows n = 1..12 of table, flattened
Götz Pfeiffer, Numbers of subgroups of various families of groups

Row lengths are given by A218913.

See A243748 for a closely related sequence.

LoadPackage("sonata");
a := function(n)
  return Length(Unique(Apply(Subgroups(SymmetricGroup(n)), Size)));
end;; # Charles R Greathouse IV, Nov 28 2016

A218915 Number of missing subgroup orders of the symmetric group, that is, i divides Factorial(n) but the symmetric group on n points does not have a subgroup of order i.

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 9, 29, 47, 86, 157, 401, 576, 1316

Offset: 0

Liam Naughton, Nov 09 2012

L. Naughton and G. Pfeiffer, Integer sequences realized by the subgroup pattern of the symmetric group, arXiv:1211.1911 [math.GR], 2012-2013.
Liam Naughton, CountingSubgroups.g
Liam Naughton and Goetz Pfeiffer, Tomlib, The GAP table of marks library

Size(Difference(DivisorsInt(Factorial(n)), DuplicateFreeList(List(ConjugacyClassesSubgroups(SymmetricGroup(n)), x->Size(Representative(x))))));

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A027423 = A@027423;
A218913 = A@218913;
a[n_] := A027423[[n+1]] - A218913[[n+1]];
a /@ Range[0, 13] (* Jean-François Alcover, Jan 08 2020 *)

a(n) = A027423(n) - A218913(n).

cp's OEIS Frontend

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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A277566 Irregular table by rows: Orders of subgroups of the symmetric group S_n.

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GAP

A218915 Number of missing subgroup orders of the symmetric group, that is, i divides Factorial(n) but the symmetric group on n points does not have a subgroup of order i.

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GAP

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