A071605
Number of ordered pairs (a,b) of elements of the symmetric group S_n such that the pair a,b generates S_n.
Original entry on oeis.org
1, 3, 18, 216, 6840, 228960, 15573600, 994533120, 85232891520, 8641918252800, 1068888956889600, 155398203460684800, 26564263279602048000
Offset: 1
Sharon Sela (sharonsela(AT)hotmail.com), Jun 02 2002
- 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.
-
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
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
A261750
Number of conjugacy classes of two-element generating sets in the symmetric group S_n.
Original entry on oeis.org
0, 1, 2, 5, 31, 163, 1576
Offset: 1
-
# GAP 4.7 code for calculating the number of distinct 2-generating sets of
# symmetric groups.
# This code is written for readability, and to minimize package dependencies.
# 2015 Attila Egri-Nagy
# decides whether the given generating sets generate the symmetric group of
# degree n or not
IsSn := function(gens,n)
return Size(Group(gens))=Factorial(n);
end;
# returns all degree n permutations (i.e., elements of the symmetric group)
AllPermsDegn := function(n)
return AsList(SymmetricGroup(IsPermGroup,n));
end;
# first 5 entries of A001691 calculated in an inefficient manner
# taking all sets of cardinality 2 and check
gensets := List([1..5],
x->Filtered(Combinations(AllPermsDegn(x),2),
y->IsSn(y,x)));
Display(List(gensets,Size));
# returns the conjugacy class representative of P under G
# calculates the conjugacy class of P and returns the minimum element
# P - set of permutations
# G - permutation group
ConjClRep := function(P, G)
return Minimum(Set(AsList(G), x-> Set(P, y->y^x)));
end;
Display(List([1..5],
x->Size(Set(gensets[x],
y->ConjClRep(y,SymmetricGroup(IsPermGroup,x))))));
Showing 1-3 of 3 results.
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