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A261750 Number of conjugacy classes of two-element generating sets in the symmetric group S_n. 0
0, 1, 2, 5, 31, 163, 1576 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Two generating sets are considered to be the same if they differ only by some relabeling of the points, i.e., conjugating by some element of S_n. For instance, the generating set {(1,2), (1,2,3,4)} is the same as {(2,3),(1,2,3,4)} by the relabeling 1->2, 2->3, 3->4, 4->1. As a non-example, the generating sets {(1,2),(1,2,3,4,5)} and {(1,3),(1,2,3,4,5)} are different, since the points in the transpositions are differently placed in the 5-cycle.

LINKS

Table of n, a(n) for n=1..7.

PROG

(GAP)

# 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))))));

CROSSREFS

Cf. A001691.

Sequence in context: A215168 A266478 A107389 * A189559 A077483 A119242

Adjacent sequences:  A261747 A261748 A261749 * A261751 A261752 A261753

KEYWORD

nonn,hard,more

AUTHOR

Attila Egri-Nagy, Aug 30 2015

STATUS

approved

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Last modified September 21 13:41 EDT 2021. Contains 347598 sequences. (Running on oeis4.)