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%I #20 Dec 23 2023 14:30:25
%S 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,
%T 1,75,40,255,36,280,255,10,36,150,0,45,50,36,90,0,30,0,0,30,12,10,0,0,
%U 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
%N 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!.
%C 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.
%e 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).
%e Triangle begins:
%e 1;
%e 1,1;
%e 1,3,1,1;
%e 1,9,4,7,4,3,1,1;
%e 1,25,10,35,6,30,15,6,15,0,6,5,0,0,1,1;
%e ...
%o (GAP)
%o # GAP 4
%o LoadPackage("SONATA") ;;
%o Print("\n") ;
%o N := Factorial(7) ;; # adjusted to the maximum n below
%o subS := EmptyPlist(N) ;;
%o for n in [1..7] do
%o for e in [1..N] do
%o subS[e] := 0 ;
%o od;
%o g := SymmetricGroup(n) ;
%o sg := Size(g) ;
%o alls := Subgroups(g) ;
%o for s in alls do
%o o := Size(s) ;
%o if o <= N then
%o subS[o] := subS[o]+1 ;;
%o fi;
%o od ;
%o for d in [1..N] do
%o if ( sg mod d ) = 0 then
%o Print(subS[d],",") ;
%o fi;
%o od;
%o Print("\n") ;
%o od;
%Y Cf. A005432 (row sums), A001189 (column d=2), A027423 (row lengths), A218913, A277566, A284210.
%K nonn,tabf
%O 1,5
%A _R. J. Mathar_, Jun 09 2014
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