login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A071605 Number of ordered pairs (a,b) of elements of the symmetric group S_n such that the pair a,b generates S_n. 7
1, 3, 18, 216, 6840, 228960, 15573600, 994533120, 85232891520, 8641918252800, 1068888956889600, 155398203460684800, 26564263279602048000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..13.

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. Luczak 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.

FORMULA

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

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

PROG

(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

CROSSREFS

Cf. A040175, A135474.

Sequence in context: A132727 A111841 A279233 * A340336 A222686 A274271

Adjacent sequences:  A071602 A071603 A071604 * A071606 A071607 A071608

KEYWORD

nonn,more,nice

AUTHOR

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

EXTENSIONS

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 1 11:38 EST 2021. Contains 349429 sequences. (Running on oeis4.)