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

Author

R. J. Mathar, Jun 09 2014

Keywords

• nonn
• tabf

Comments

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.

Examples

			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;
  ...
		

Crossrefs

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

Programs

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;

Extensions

Edited by Peter Munn, Mar 06 2025