Original entry on oeis.org
1, 3, 9, 36, 285, 1908, 21630, 197328, 2113911, 23814810, 294557142, 3893052636, 55457566905
Offset: 1
A040175
a(n) = n! times probability that an ordered pair of elements of S_n chosen at random (with replacement) generate S_n.
Original entry on oeis.org
3, 9, 57, 318, 3090, 24666, 234879, 2381481, 26777922, 324421053, 4265966685
Offset: 3
Probabilities for n=1,2,3,... are 1, 3/4, 1/2, 3/8, 19/40, ...
- 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.
- 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.
- 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.
A001691
Number of two-element generating sets in the symmetric group S_n.
Original entry on oeis.org
0, 1, 9, 108, 3420, 114480, 7786800, 497266560, 42616445760, 4320959126400, 534444478444800, 77699101730342400, 13282131639801024000
Offset: 1
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
-
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] * Binomial(orders[k],2);
fi;
od;
return num;
end; # Stephen A. Silver, Feb 20 2013
A220754
Number of ordered triples (a,b,c) of elements of the symmetric group S_n such that the triple a,b,c generates a transitive group.
Original entry on oeis.org
1, 7, 194, 12858, 1647384, 361351560, 125116670160, 64439768489040, 47159227114392960, 47285264408385951360, 63057420721939066617600, 109118766834521171299756800, 239996135160204867851157273600, 659114500480471292127627441484800
Offset: 1
-
nn=14; b=Sum[n!^3 x^n/n!,{n,0,nn}]; Drop[Range[0,nn]!CoefficientList[Series[Log[b],{x,0,nn}],x],1]
-
N = 66; x = 'x + O('x^N);
egf = log(sum(n=0, N, n!^2*x^n));
gf = serlaplace(egf);
v = Vec(gf)
/* Joerg Arndt, Apr 14 2013 */
A086373
Number of ordered triples (a,b,c) of elements of the symmetric group S_n such that a,b,c generate S_n.
Original entry on oeis.org
1, 7, 168, 10080, 1401120, 303730560, 109469465280, 56335746378240, 41263790481123840, 41372254858231987200, 55175243131277553715200, 95478523289749232323891200, 209996618265179127555767193600
Offset: 1
Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 06 2003
-
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]^3;
fi;
od;
return num;
end; # Stephen A. Silver, Feb 20 2013
A224539
Number of pairs of elements of the full transformational monoid T_n on {1,...,n} which generate a synchronizing monoid.
Original entry on oeis.org
1, 12, 549, 51520, 8063385, 1871446896
Offset: 1
- P. J. Cameron, Dixon's theorem and random synchronization, Discrete Math., 313 (2013), 1233-1246. (a(1)-a(6) were computed by James Mitchell.)
A086502
Number of ordered pairs (a,b) of elements of the symmetric group S_n such that the pair a,b generates a proper subgroup of S_n.
Original entry on oeis.org
0, 1, 18, 360, 7560, 289440, 9828000, 631169280, 46449002880, 4526271187200, 524461965350400, 74044329341875200, 12211524764030592000
Offset: 1
Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 09 2003
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