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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!.

%I A243748 #23 Mar 06 2025 09:30:10
%S A243748 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 A243748 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 A243748 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 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!.
%C A243748 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.
%e A243748 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 A243748 Triangle begins:
%e A243748   1;
%e A243748   1,1;
%e A243748   1,3,1,1;
%e A243748   1,9,4,7,4,3,1,1;
%e A243748   1,25,10,35,6,30,15,6,15,0,6,5,0,0,1,1;
%e A243748   ...
%o A243748 (GAP)
%o A243748 # GAP 4
%o A243748 LoadPackage("SONATA") ;;
%o A243748 Print("\n") ;
%o A243748 N := Factorial(7) ;; # adjusted to the maximum n below
%o A243748 subS := EmptyPlist(N) ;;
%o A243748 for n in [1..7] do
%o A243748     for e in [1..N] do
%o A243748         subS[e] := 0 ;
%o A243748     od;
%o A243748     g := SymmetricGroup(n) ;
%o A243748     sg := Size(g) ;
%o A243748     alls := Subgroups(g) ;
%o A243748     for s in alls do
%o A243748         o := Size(s) ;
%o A243748         if o <= N then
%o A243748             subS[o] := subS[o]+1 ;;
%o A243748         fi;
%o A243748     od ;
%o A243748     for d in [1..N] do
%o A243748         if ( sg mod d ) = 0 then
%o A243748             Print(subS[d],",") ;
%o A243748         fi;
%o A243748     od;
%o A243748     Print("\n") ;
%o A243748 od;
%Y A243748 Cf. A005432 (row sums), A001189 (column d=2), A027423 (row lengths), A218913, A277566, A284210.
%K A243748 nonn,tabf
%O A243748 1,5
%A A243748 _R. J. Mathar_, Jun 09 2014
%E A243748 Edited by _Peter Munn_, Mar 06 2025