cp's OEIS Frontend

A071605 Number of ordered pairs (a,b) of elements of the symmetric group S_n such that the pair a,b generates S_n.

%I A071605 #28 Jun 22 2025 13:43:07
%S A071605 1,3,18,216,6840,228960,15573600,994533120,85232891520,8641918252800,
%T A071605 1068888956889600,155398203460684800,26564263279602048000
%N A071605 Number of ordered pairs (a,b) of elements of the symmetric group S_n such that the pair a,b generates S_n.
%C A071605 a(n) is an Eulerian function of S_n. - _Kenneth G. Hawes_, Nov 25 2019
%H A071605 L. Babai, <a href="http://dx.doi.org/10.1016/0097-3165(89)90068-X">The probability of generating the symmetric group</a>, J. Combin. Theory, A52 (1989), 148-153.
%H A071605 J. D. Dixon, <a href="http://dx.doi.org/10.1007/BF01110210">The probability of generating the symmetric group</a>, Math. Z. 110 (1969) 199-205.
%H A071605 J. D. Dixon, <a href="http://dx.doi.org/10.1016/j.disc.2007.07.021">Problem 923 (BCC20.17), Indecomposable permutations and transitive groups</a>, in Research Problems from the 20th British Combinatorial Conference, Discrete Math., 308 (2008), 621-630.
%H A071605 P. Hall, <a href="https://doi.org/10.1093/qmath/os-7.1.134">The Eulerian functions of a group</a>, Quart. J. Math. 7 (1936), 134-151.
%H A071605 T. Łuczak and L. Pyber, <a href="http://dx.doi.org/10.1017/S0963548300000869">On random generation of the symmetric group</a>, Combin. Probab. Comput., 2 (1993), 505-512.
%H A071605 A. Maroti and C. M. Tamburini, <a href="http://dx.doi.org/10.1007/s00013-010-0216-z">Bounds for the probability of generating the symmetric and alternating groups</a>, Arch. Math. (Basel), 96 (2011), 115-121.
%F A071605 Except for n=2 (because of the "replacement") in A040175, a(n) = n! * A040175(n).
%F A071605 a(n) = 2 * A001691(n) for n > 2.
%o A071605 (GAP)
%o A071605 a := function(n)
%o A071605   local tom, mu, lens, orders, num, k;
%o A071605   tom := TableOfMarks(Concatenation("S",String(n)));
%o A071605   if tom = fail then tom := TableOfMarks(SymmetricGroup(n)); fi;
%o A071605   mu :=  MoebiusTom(tom).mu;
%o A071605   lens := LengthsTom(tom);
%o A071605   orders := OrdersTom(tom);
%o A071605   num := 0;
%o A071605   for k in [1 .. Length(lens)] do
%o A071605     if IsBound(mu[k]) then
%o A071605       num := num + mu[k] * lens[k] * orders[k]^2;
%o A071605     fi;
%o A071605   od;
%o A071605   return num;
%o A071605 end; # _Stephen A. Silver_, Feb 20 2013
%Y A071605 Cf. A001691, A040175, A135474.
%K A071605 nonn,more,nice
%O A071605 1,2
%A A071605 Sharon Sela (sharonsela(AT)hotmail.com), Jun 02 2002
%E A071605 a(10)-a(13) added by _Stephen A. Silver_, Feb 20 2013