A071605 - OEIS

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

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OFFSET

1,2

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, 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.

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

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