A243748 Triangle T(n,d): the number of subgroups of order d of the symmetric group S_n as d runs through the divisors 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

1,5

Comments

The columns skip the subgroups of S_n which are known not to exist (because their order does not divide the order of S_n, n!). This is just a reduction of rows in the triangle by omitting a large number of zeros.

Links

Table of n, a(n) for n=1..86.

Example

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;
  ...

Prog

(GAP)
# 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;

Crossrefs

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

Sequence in context: A366787 A348038 A327564 * A340149 A340075 A307847

Adjacent sequences: A243745 A243746 A243747 * A243749 A243750 A243751

Keyword

nonn,tabf

Author

R. J. Mathar, Jun 09 2014

Status

approved