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A001691 Number of two-element generating sets in the symmetric group S_n.

%I A001691 M4660 N1995 #35 Feb 01 2022 07:12:34
%S A001691 0,1,9,108,3420,114480,7786800,497266560,42616445760,4320959126400,
%T A001691 534444478444800,77699101730342400,13282131639801024000
%N A001691 Number of two-element generating sets in the symmetric group S_n.
%D A001691 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001691 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001691 J. Denes, <a href="http://dx.doi.org/10.1016/S0021-9800(70)80017-5">Some combinatorial properties of transformations and their connections with the theory of graphs</a>, J. Combin. Theory, 9 (1970), 108-116.
%F A001691 a(n) = A071605(n)/2 for n > 2.
%o A001691 (GAP)
%o A001691 a := function(n)
%o A001691   local tom, mu, lens, orders, num, k;
%o A001691   tom := TableOfMarks(Concatenation("S",String(n)));
%o A001691   if tom = fail then tom := TableOfMarks(SymmetricGroup(n)); fi;
%o A001691   mu :=  MoebiusTom(tom).mu;
%o A001691   lens := LengthsTom(tom);
%o A001691   orders := OrdersTom(tom);
%o A001691   num := 0;
%o A001691   for k in [1 .. Length(lens)] do
%o A001691     if IsBound(mu[k]) then
%o A001691       num := num + mu[k] * lens[k] * Binomial(orders[k],2);
%o A001691     fi;
%o A001691   od;
%o A001691   return num;
%o A001691 end; # _Stephen A. Silver_, Feb 20 2013
%Y A001691 Cf. A071605, A086373.
%K A001691 nonn,more
%O A001691 1,3
%A A001691 _N. J. A. Sloane_
%E A001691 a(8)-a(9) (derived from A071605) added by _Stephen A. Silver_, Feb 17 2013
%E A001691 a(10)-a(13) added by _Stephen A. Silver_, Feb 20 2013