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 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!. 2
 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 (list; graph; refs; listen; history; text; internal format)
 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 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). 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 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: A229142 A156535 A327564 * A340149 A340075 A307847 Adjacent sequences:  A243745 A243746 A243747 * A243749 A243750 A243751 KEYWORD nonn,tabf AUTHOR R. J. Mathar, Jun 09 2014 STATUS approved

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Last modified April 22 12:53 EDT 2021. Contains 343177 sequences. (Running on oeis4.)