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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 (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 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

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