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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
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, 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
Sequence in context: A132727 A111841 A279233 * A340336 A222686 A274271
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