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

1, 3, 18, 216, 6840, 228960, 15573600, 994533120, 85232891520, 8641918252800, 1068888956889600, 155398203460684800, 26564263279602048000

Offset: 1

Author

Sharon Sela (sharonsela(AT)hotmail.com), Jun 02 2002

Keywords

• nonn
• more
• nice

Comments

a(n) is an Eulerian function of S_n. - Kenneth G. Hawes, Nov 25 2019

Links

• L. Babai, The probability of generating the symmetric group, J. Combin. Theory, A52 (1989), 148-153.
• J. D. Dixon, The probability of generating the symmetric group, Math. Z. 110 (1969) 199-205.
• J. D. Dixon, Problem 923 (BCC20.17), Indecomposable permutations and transitive groups, in Research Problems from the 20th British Combinatorial Conference, Discrete Math., 308 (2008), 621-630.
• P. Hall, The Eulerian functions of a group, Quart. J. Math. 7 (1936), 134-151.
• T. Łuczak and L. Pyber, On random generation of the symmetric group, Combin. Probab. Comput., 2 (1993), 505-512.
• A. Maroti and C. M. Tamburini, Bounds for the probability of generating the symmetric and alternating groups, Arch. Math. (Basel), 96 (2011), 115-121.

Crossrefs

Cf. A001691, A040175, A135474.

Programs

GAP

a := function(n)
  local tom, mu, lens, orders, num, k;
  tom := TableOfMarks(Concatenation("S",String(n)));
  if tom = fail then tom := TableOfMarks(SymmetricGroup(n)); fi;
  mu :=  MoebiusTom(tom).mu;
  lens := LengthsTom(tom);
  orders := OrdersTom(tom);
  num := 0;
  for k in [1 .. Length(lens)] do
    if IsBound(mu[k]) then
      num := num + mu[k] * lens[k] * orders[k]^2;
    fi;
  od;
  return num;
end; # Stephen A. Silver, Feb 20 2013

Formula

Except for n=2 (because of the "replacement") in A040175, a(n) = n! * A040175(n).

a(n) = 2 * A001691(n) for n > 2.

Extensions

a(10)-a(13) added by Stephen A. Silver, Feb 20 2013