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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 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.)