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